Search: a103210 -id:a103210
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1, 3, 21, 162, 1365, 12219, 114156, 1100649, 10871175, 109438830, 1118798079, 11583712617, 121219182504, 1280065637487, 13623341795049, 145977237305874, 1573536198376401, 17051418418204671, 185646639499541892
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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G.f.: (1-3x+x^2-sqrt(1-14x+27x^2-14x^3+x^4))/(4x);
G.f.: 1/(1-3x/((1-x)^2-2x/(1-3x/((1-x)^2-2x/(1-3x/((1-x)^2-2x/(1-3x/(1-... (continued fraction);
a(n) = Sum_{k=0..n} (0^(n+k)+C(n+k-1,2k-1))*A103210(k) = 0^n + Sum_{k=0..n} C(n+k-1,2k-1)*A103210(k).
Conjecture: (n+1)*a(n) +7*(-2*n+1)*a(n-1) +27*(n-2)*a(n-2) +7*(-2*n+7)*a(n-3) +(n-5)*a(n-4)=0. - R. J. Mathar, Feb 10 2015
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MAPLE
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if n = 0 then
1;
else
add((0^(n+k)+binomial(n+k-1, 2*k-1))*A103210(k), k=0..n) ;
end if;
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MATHEMATICA
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CoefficientList[Series[(1 - 3*t + t^2 - Sqrt[1 - 14*t + 27*t^2 - 14*t^3 + t^4])/(4*t), {t, 0, 50}], t] (* G. C. Greubel, May 23 2016 *)
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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1, 4, 25, 187, 1552, 13771, 127927, 1228576, 12099751, 121538581, 1240336660, 12824049277, 134043231781, 1414108869268, 15037450664317, 161014687970191, 1734550886346592, 18785969304551263, 204432608804093155
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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G.f.: (1-3x+x^2-sqrt(1-14x+27x^2-14x^3+x^4))/(4x(1-x));
G.f.: 1/(1-x-3x/(1-x-2x/(1-x-3x/(1-x-2x/(1-x-3x/(1-x-2x/(1-.... (continued fraction);
a(n) = Sum_{k=0..n} C(n+k,2k)*A103210(k).
Conjecture: (n+1)*a(n) + 3*(-5*n+2)*a(n-1) + (41*n-61)*a(n-2) + (-41*n+103)*a(n-3) + 3*(5*n-18)*a(n-4) + (-n+5)*a(n-5) = 0. - R. J. Mathar, Feb 10 2015
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MAPLE
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MATHEMATICA
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CoefficientList[Series[(1 - 3*t + t^2 - Sqrt[1 - 14*t + 27*t^2 - 14*t^3 + t^4])/(4*t*(1 - t)), {t, 0, 50}], t] (* G. C. Greubel, May 23 2016 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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A090181
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Triangle of Narayana (A001263) with 0 <= k <= n, read by rows.
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+10
41
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1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1, 0, 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 0, 1, 55, 825, 4950, 13860
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OFFSET
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0,9
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COMMENTS
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Number of Dyck n-paths with exactly k peaks. - Peter Luschny, May 10 2014
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LINKS
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FORMULA
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Triangle T(n, k), read by rows, given by [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938. T(0, 0) = 1, T(n, 0) = 0 for n>0, T(n, k) = C(n-1, k-1)*C(n, k-1)/k for k>0.
Coefficient array of the polynomials P(n,x) = x^n*2F1(-n,-n+1;2;1/x).
T(n,k) = Sum_{j=0..n} (-1)^(j-k)*C(2n-j,j)*C(j,k)*A000108(n-j). (End)
T(n, k) = C(n,n-k)*C(n-1,n-k)/(n-k+1). - Peter Luschny, May 10 2014
E.g.f.: 1+Integral((sqrt(t)*exp((1+t)*x)*BesselI(1,2*sqrt(t)*x))/x dx). - Peter Luschny, Oct 30 2014
T(n, k) = [x^k] (((2*n - 1)*(1 + x)*p(n-1, x) - (n - 2)*(x - 1)^2*p(n-2, x))/(n + 1)) with p(0, x) = 1 and p(1, x) = x. - Peter Luschny, Apr 26 2022
Recursion based on rows (see the Python program):
T(n, k) = (((B(k) + B(k-1))*(2*n - 1) - (A(k) - 2*A(k-1) + A(k-2))*(n-2))/(n+1)), where A(k) = T(n-2, k) and B(k) = T(n-1, k), for n >= 3. # Peter Luschny, May 02 2022
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EXAMPLE
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Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1, 1;
[3] 0, 1, 3, 1;
[4] 0, 1, 6, 6, 1;
[5] 0, 1, 10, 20, 10, 1;
[6] 0, 1, 15, 50, 50, 15, 1;
[7] 0, 1, 21, 105, 175, 105, 21, 1;
[8] 0, 1, 28, 196, 490, 490, 196, 28, 1;
[9] 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1;
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MAPLE
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A090181 := (n, k) -> binomial(n, n-k)*binomial(n-1, n-k)/(n-k+1): seq(print( seq(A090181(n, k), k=0..n)), n=0..5); # Peter Luschny, May 10 2014
# Alternatively:
egf := 1+int((sqrt(t)*exp((1+t)*x)*BesselI(1, 2*sqrt(t)*x))/x, x);
s := n -> n!*coeff(series(egf, x, n+2), x, n); seq(print(seq(coeff(s(n), t, j), j=0..n)), n=0..9); # Peter Luschny, Oct 30 2014
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MATHEMATICA
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Flatten[Table[Sum[(-1)^(j-k) * Binomial[2n-j, j] * Binomial[j, k] * CatalanNumber[n-j], {j, 0, n}], {n, 0, 11}, {k, 0, n}]] (* Indranil Ghosh, Mar 05 2017 *)
p[0, _] := 1; p[1, x_] := x; p[n_, x_] := ((2 n - 1) (1 + x) p[n - 1, x] - (n - 2) (x - 1)^2 p[n - 2, x]) / (n + 1);
Table[CoefficientList[p[n, x], x], {n, 0, 9}] // TableForm (* Peter Luschny, Apr 26 2022 *)
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PROG
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(Sage)
U = [0]*(n+1)
for d in DyckWords(n):
U[d.number_of_peaks()] +=1
return U
(Python) from functools import cache
@cache
def Trow(n):
if n == 0: return [1]
if n == 1: return [0, 1]
if n == 2: return [0, 1, 1]
A = Trow(n - 2) + [0, 0]
B = Trow(n - 1) + [1]
for k in range(n - 1, 1, -1):
B[k] = (((B[k] + B[k - 1]) * (2 * n - 1)
- (A[k] - 2 * A[k - 1] + A[k - 2]) * (n - 2)) // (n + 1))
return B
(PARI)
c(n) = binomial(2*n, n)/ (n+1);
tabl(nn) = {for(n=0, nn, for(k=0, n, print1(sum(j=0, n, (-1)^(j-k) * binomial(2*n-j, j) * binomial(j, k) * c(n-j)), ", "); ); print(); ); };
(Magma) [[(&+[(-1)^(j-k)*Binomial(2*n-j, j)*Binomial(j, k)*Binomial(2*n-2*j, n-j)/(n-j+1): j in [0..n]]): k in [0..n]]: n in [0..10]];
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CROSSREFS
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Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n) for x=0,1,2,3,4,5,6,7,8,9. - Philippe Deléham, Aug 10 2006
Sum_{k=0..n} x^(n-k)*T(n,k) = A090192(n+1), A000012(n), A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. - Philippe Deléham, Oct 21 2006
Sum_{k=0..n} T(n,k)*x^k*(x-1)^(n-k) = A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Oct 20 2007
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KEYWORD
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AUTHOR
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STATUS
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approved
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A088617
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Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.
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+10
38
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1, 1, 1, 1, 3, 2, 1, 6, 10, 5, 1, 10, 30, 35, 14, 1, 15, 70, 140, 126, 42, 1, 21, 140, 420, 630, 462, 132, 1, 28, 252, 1050, 2310, 2772, 1716, 429, 1, 36, 420, 2310, 6930, 12012, 12012, 6435, 1430, 1, 45, 660, 4620, 18018, 42042, 60060, 51480, 24310, 4862
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OFFSET
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0,5
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COMMENTS
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Row sums: A006318 (Schroeder numbers). Essentially same as triangle A060693 transposed.
T(n,k) is number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k U's. E.g., T(2,1)=3 because we have UHD, UDH and HUD. - Emeric Deutsch, Dec 06 2003
Conjecture: The expected number of U's in a Schroeder n-path is asymptotically Sqrt[1/2]*n for large n. - David Callan, Jul 25 2008
T(n, k) is also the number of order-preserving and order-decreasing partial transformations (of an n-chain) of width k (width(alpha) = |Dom(alpha)|). - Abdullahi Umar, Oct 02 2008
The antidiagonals of this lower triangular matrix are the rows of A055151. - Tom Copeland, Jun 17 2015
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REFERENCES
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Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 449.
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LINKS
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FORMULA
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Triangle T(n, k) read by rows; given by [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
Sum_{k=0..n} T(n, k)*x^k*(1-x)^(n-k) = A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. - Philippe Deléham, Aug 18 2005
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A107841(n), A080243(n), A000007(n), A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Oct 18 2007
O.g.f. (with initial 1 excluded) is the series reversion with respect to x of (1-t*x)*x/(1+x). Cf. A062991 and A089434. - Peter Bala, Jul 31 2012
G.f.: 1 + (1 - x - T(0))/y, where T(k) = 1 - x*(1+y)/( 1 - x*y/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 03 2013
O.g.f. A(x,t) = ( 1 - x - sqrt((1 - x)^2 - 4*x*t) )/(2*x*t) = 1 + (1 + t)*x + (1 + 3*t + 2*t^2)*x^2 + ....
1 + x*(dA(x,t)/dx)/A(x,t) = 1 + (1 + t)*x + (1 + 4*t + 3*t^2)*x^2 + ... is the o.g.f. for A123160.
For n >= 1, the n-th row polynomial equals (1 + t)/(n+1)*Jacobi_P(n-1,1,1,2*t+1). Removing a factor of 1 + t from the row polynomials gives the row polynomials of A033282. (End)
The o.g.f. G(x,t) = {1 - (2t+1) x - sqrt[1 - (2t+1) 2x + x^2]}/2x = (t + t^2) x + (t + 3t^2 + 2t^3) x^2 + (t + 6t^2 + 10t^3 + 5t^3) x^3 + ... generating shifted rows of this entry, excluding the first, was given in my 2008 formulas for A033282 with an o.g.f. f1(x,t) = G(x,t)/(1+t) for A033282. Simple transformations presented there of f1(x,t) are related to A060693 and A001263, the Narayana numbers. See also A086810.
The inverse of G(x,t) is essentially given in A033282 by x1, the inverse of f1(x,t): Ginv(x,t) = x [1/(t+x) - 1/(1+t+x)] = [((1+t) - t) / (t(1+t))] x - [((1+t)^2 - t^2) / (t(1+t))^2] x^2 + [((1+t)^3 - t^3) / (t(1+t))^3] x^3 - ... . The coefficients in t of Ginv(xt,t) are the o.g.f.s of the diagonals of the Pascal triangle A007318 with signed rows and an extra initial column of ones. The numerators give the row o.g.f.s of signed A074909.
(End)
T(n, k) = [x^k] hypergeom([-n, 1 + n], [2], -x). - Peter Luschny, Apr 26 2022
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EXAMPLE
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Triangle begins:
[0] 1;
[1] 1, 1;
[2] 1, 3, 2;
[3] 1, 6, 10, 5;
[4] 1, 10, 30, 35, 14;
[5] 1, 15, 70, 140, 126, 42;
[6] 1, 21, 140, 420, 630, 462, 132;
[7] 1, 28, 252, 1050, 2310, 2772, 1716, 429;
[8] 1, 36, 420, 2310, 6930, 12012, 12012, 6435, 1430;
[9] 1, 45, 660, 4620, 18018, 42042, 60060, 51480, 24310, 4862;
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MAPLE
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R := n -> simplify(hypergeom([-n, n + 1], [2], -x)):
Trow := n -> seq(coeff(R(n, x), x, k), k = 0..n):
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MATHEMATICA
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Table[Binomial[n+k, n] Binomial[n, k]/(k+1), {n, 0, 10}, {k, 0, n}]//Flatten (* Michael De Vlieger, Aug 10 2017 *)
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PROG
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(PARI) {T(n, k)= if(k+1, binomial(n+k, n)*binomial(n, k)/(k+1))}
(Magma) [[Binomial(n+k, n)*Binomial(n, k)/(k+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jun 18 2015
(SageMath) flatten([[binomial(n+k, 2*k)*catalan_number(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 22 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A060693
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Triangle (0 <= k <= n) read by rows: T(n, k) is the number of Schröder paths from (0,0) to (2n,0) having k peaks.
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+10
25
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1, 1, 1, 2, 3, 1, 5, 10, 6, 1, 14, 35, 30, 10, 1, 42, 126, 140, 70, 15, 1, 132, 462, 630, 420, 140, 21, 1, 429, 1716, 2772, 2310, 1050, 252, 28, 1, 1430, 6435, 12012, 12012, 6930, 2310, 420, 36, 1, 4862, 24310, 51480, 60060, 42042, 18018, 4620, 660, 45, 1, 16796
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OFFSET
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0,4
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COMMENTS
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The rows sum to A006318 (Schroeder numbers), the left column is A000108 (Catalan numbers); the next-to-left column is A001700, the alternating sum in each row but the first is 0.
T(n,k) is the number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k peaks. Example: T(2,1)=3 because we have UU*DD, U*DH and HU*D, the peaks being shown by *. E.g., T(n,k) = binomial(n,k)*binomial(2n-k,n-1)/n for n>0. - Emeric Deutsch, Dec 06 2003
T(n,k) is also the number of rooted plane trees with maximal degree 3 and k vertices of degree 2 (a node may have at most 2 children, and there are exactly k nodes with 1 child). Equivalently, T(n,k) is the number of syntactically different expressions that can be formed that use a unary operation k times, a binary operation n-k times, and nothing else (sequence of operands is fixed). - Lars Hellstrom (Lars.Hellstrom(AT)residenset.net), Dec 08 2009
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LINKS
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FORMULA
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Triangle T(n, k) (0 <= k <= n) read by rows; given by [1, 1, 1, 1, 1, ...] DELTA [1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 12 2003
If C_n(x) is the g.f. of row n of the Narayana numbers (A001263), C_n(x) = Sum_{k=1..n} binomial(n,k-1)*(binomial(n-1,k-1)/k) * x^k and T_n(x) is the g.f. of row n of T(n,k), then T_n(x) = C_n(x+1), or T(n,k) = [x^n]Sum_{k=1..n}(A001263(n,k)*(x+1)^k). - Mitch Harris, Jan 16 2007, Jan 31 2007
G.f.: (1 - t*y - sqrt((1-y*t)^2 - 4*y)) / 2.
T(n, k) = binomial(2n-k, n)*binomial(n, k)/(n-k+1). - Philippe Deléham, Dec 07 2003
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n), for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Apr 01 2007
Sum_{k=0..n} T(n,k)*x^(n-k) = (-1)^n*A107841(n), A080243(n), A000007(n), A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Oct 18 2007
G.f.: 1/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x/(1-.... (continued fraction);
G.f.: 1/(1-(x+xy)/(1-x/(1-(x+xy)/(1-x/(1-(x+xy)/(1-.... (continued fraction). (End)
T(n,k) = [k<=n]*(Sum_{j=0..n} binomial(n,j)^2*binomial(j,k))/(n-k+1). - Paul Barry, May 28 2009
With F(x,t) = (1-(2+t)*x-sqrt(1-2*(2+t)*x+(t*x)^2))/(2*x) an o.g.f. (nulling the n=0 term) in x for the A060693 polynomials in t,
G(x,t) = x/(1+t+(2+t)*x+x^2) is the compositional inverse in x.
Consequently, with H(x,t) = 1/(dG(x,t)/dx) = (1+t+(2+t)*x+x^2)^2 / (1+t-x^2), the n-th A060693 polynomial in t is given by (1/n!)*((H(x,t)*d/dx)^n) x evaluated at x=0, i.e., F(x,t) = exp(x*H(u,t)*d/d) u, evaluated at u = 0.
Also, dF(x,t)/dx = H(F(x,t),t). (End)
Rows of this entry are non-vanishing antidiagonals of A097610. See p. 14 of Agapito et al. for a bivariate generating function and its inverse. - Tom Copeland, Feb 03 2016
Sum_{k=0..n} (-1)^k*(1+x*(n-k))*T(n,k) = x + (1-x)*A000007(n).
(End)
Conjecture: Sum_{k=0..n} (-1)^k*T(n,k)*(n+1-k)^2 = 1+n+n^2. - Werner Schulte, Jan 11 2017
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EXAMPLE
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Triangle begins:
00: [ 1]
01: [ 1, 1]
02: [ 2, 3, 1]
03: [ 5, 10, 6, 1]
04: [ 14, 35, 30, 10, 1]
05: [ 42, 126, 140, 70, 15, 1]
06: [ 132, 462, 630, 420, 140, 21, 1]
07: [ 429, 1716, 2772, 2310, 1050, 252, 28, 1]
08: [ 1430, 6435, 12012, 12012, 6930, 2310, 420, 36, 1]
09: [ 4862, 24310, 51480, 60060, 42042, 18018, 4620, 660, 45, 1]
10: [16796, 92378, 218790, 291720, 240240, 126126, 42042, 8580, 990, 55, 1]
...
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MAPLE
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MATHEMATICA
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t[n_, k_] := Binomial[n, k]*Binomial[2 n - k, n]/(n - k + 1); Flatten[Table[t[n, k], {n, 0, 9}, {k, 0, n}]] (* Robert G. Wilson v, May 30 2011 *)
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PROG
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(PARI) T(n, k) = binomial(n, k)*binomial(2*n - k, n)/(n - k + 1);
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print); \\ Indranil Ghosh, Jul 28 2017
(Python)
from sympy import binomial
def T(n, k): return binomial(n, k) * binomial(2 * n - k, n) / (n - k + 1)
for n in range(11): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Jul 28 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A107841
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Series reversion of x*(1-3*x)/(1-x).
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+10
22
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1, 2, 10, 62, 430, 3194, 24850, 199910, 1649350, 13879538, 118669210, 1027945934, 9002083870, 79568077034, 708911026210, 6359857112438, 57403123415350, 520895417047010, 4749381474135850, 43489017531266654, 399755692955359630, 3687437532852484442, 34121911117572911410
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OFFSET
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0,2
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COMMENTS
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In general, the series reversion of x(1-r*x)/(1-x) has g.f. (1+x-sqrt(1+2*(1-2*r)*x+x^2))/(2*r) and general term given by a(n)=(1/(n+1))sum{k=0..n, C(n+1,k)C(2n-k,n)(-1)^k*r^(n-k)}; a(n)=(1/(n+1))sum{k=0..n, C(n+1,k+1)C(n+k,k)(-1)^(n-k)*r^k}; a(n)=sum{k=0..n, (1/(k+1))*C(n,k)C(n+k,k)(-1)^(n-k)*r^k}; a(n)=sum{k=0..n, A088617(n,k)*(-1)^(n-k)*r^k}.
The Hankel transform of this sequence is 6^C(n+1,2). - Philippe Deléham, Oct 29 2007
Number of Dyck n-paths with three colors of up (U,a,b) and one color of down (D) avoiding UD. - David Scambler, Jun 24 2013
This sequence is implied in the turbulence solutions of the incompressible Navier-Stokes equations in R^3. a(n) = numbers of realizable vorticity eddies in terms of initial conditions. - Fung Lam, Dec 31 2013
Conjugate sequence to this series is defined by series reversion of x(1+3*x)/(1+x), G.f.: ((x-1)-sqrt(1-10*x+ x^2))/(6*x). Conjugate sequence is the negation of this series except a(0). - Fung Lam, Jan 16 2014
Complete Chebyshev transform is G.f. = 3*F((1-x^2)/(1+x^2)), where F(x) is the g.f. of A107841. Real part of G.f. (= (1 - sqrt(3*x^4-2))/((1+x^2))) generates periodic sequence A056594. In general, for reversion of x*(1-r*x)/(1-x), r>=2, Real part of r*F((1-x^2)/(1+x^2)) (= (1 - sqrt(r*x^4 - r + 1))/(1+x^2)) generates A056594. - Fung Lam, Apr 29 2014
a(n) is the number of small Schröder n-paths with 2 types of up steps (i.e., lattice paths from (0,0) to (2n,0) using steps U1=U2=(1,1), F=(2,0), D=(1,-1), with no F steps on the x-axis). - Yu Hin Au, Dec 07 2019
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LINKS
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FORMULA
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G.f.: (1+x-sqrt(1-10x+x^2))/(6x).
a(n) = (1/(n+1))sum{k=0..n, C(n+1, k)C(2n-k, n)(-1)^k*3^(n-k)}.
a(n) = (1/(n+1))sum{k=0..n, C(n+1, k+1)C(n+k, k)(-1)^(n-k)*3^k}.
a(n) = sum{k=0..n, (1/(k+1))*C(n, k)C(n+k, k)(-1)^(n-k)*3^k}.
a(n) = sum{k=0..n, A088617(n, k)*(-1)^(n-k)*3^k}.
G.f.: 1/(1-2x/(1-3x/(1-2x/(1-3x/(1-2x/(1-3x/(1-2x/(1-3x........ (continued fraction). - Paul Barry, Dec 15 2008
G.f.: 1/(1-2x/(1-x-2x/(1-x-2x/(1-x-2x/(1-x-2x/(1-... (continued fraction).
G.f.: 1/(1-2x-6x^2/(1-5x-6x^2/(1-5x-6x^2/(1-5x-6x^2/(1-... (continued fraction). (End)
G.f.: 1/(1+x-3x/(1+x-3x/(1+x-3x/(1+x-3x/(1+x-3x/(1+... (continued fraction). - Paul Barry, Mar 18 2011
D-finite with recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(12+5*sqrt(6))*(5+2*sqrt(6))^n/(6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
a(n+1) is the coefficient of x^(n+1) in 2*sum{j,1,n}((sum{k,1,n}a(k)x^k)^(j+1)), a(1)=1 with offset by 1. - Fung Lam, Dec 31 2013
The series reversion of x*(1 - r*x)/(1 - x) is D-finite with the general recurrence n*a(n) - (2*r-1)*(2*n-3)*a(n-1) + (n-3)*a(n-2) = 0 and with initial values a(1) = 1, a(2) = r-1, a(3) = (2*r-1)*(r-1). This sequence uses r=3, cf. crossrefs. - Georg Fischer, Sep 14 2024
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MAPLE
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seq(simplify((-1)^n*hypergeom([-n, n + 1], [2], 3)), n=0..10); # Georg Fischer, Sep 14 2024 (from Peter Luschny's formula in A131763, with last parameter r=3)
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MATHEMATICA
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CoefficientList[Series[(1+x-Sqrt[1-10*x+x^2])/(6*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)
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PROG
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(PARI) x='x+O('x^66); Vec(serreverse(x*(1-3*x)/(1-x))) \\ Joerg Arndt, May 15 2013
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CROSSREFS
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KEYWORD
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easy,nonn,changed
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AUTHOR
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STATUS
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approved
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A103209
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Square array T(n,d) read by antidiagonals: number of structurally-different guillotine partitions of a d-dimensional box in R^d by n hyperplanes.
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+10
13
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1, 1, 2, 1, 6, 3, 1, 22, 15, 4, 1, 90, 93, 28, 5, 1, 394, 645, 244, 45, 6, 1, 1806, 4791, 2380, 505, 66, 7, 1, 8558, 37275, 24868, 6345, 906, 91, 8, 1, 41586, 299865, 272188, 85405, 13926, 1477, 120, 9, 1, 206098, 2474025, 3080596, 1204245, 229326, 26845
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OFFSET
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1,3
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COMMENTS
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The columns are the row sums of the inverses of the Riordan arrays ((1-d*x)/(1-x),x(1-d*x)/(1-x)), that is, of the Riordan arrays ((1+x-sqrt(1+2(1-2*d)x+x^2)/(2*d*x),(1+x-sqrt(1+2(1-2*d)x+x^2)/(2*d)). - Paul Barry, May 24 2005
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LINKS
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FORMULA
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T(n, d) = (1/n) * sum[i=0..n-1, C(n, i)*C(n, i+1)*(d-1)^i*d^(n-i) ], T(n, 0)=1.
G.f. of d-th column: [1-z-(z^2-4dz+2z+1)^(1/2)]/(2dz-2z).
T(n, k) = sum{j=0..n, C(n+j, 2j)*k^j*C(j)}, C(n) as in A000108. - Paul Barry, May 21 2005
T(n, k) = hypergeom([-n, n+1], [2], -k). - Peter Luschny, May 23 2014
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EXAMPLE
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1,...1,....1,.....1,......1,......1,.......1,.......1,.......1,
1,...2,....3,.....4,......5,......6,.......7,.......8,.......9,
1,...6,...15,....28,.....45,.....66,......91,.....120,.....153,
1,..22,...93,...244,....505,....906,....1477,....2248,....3249,
1,..90,..645,..2380,...6345,..13926,...26845,...47160,...77265,
1,.394,.4791,.24868,..85405,.229326,..522739,.1059976,.1968633,
1,1806,37275,272188,1204245,3956106,10663471,24958200,52546473,
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MAPLE
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T := (n, k) -> hypergeom([-n, n+1], [2], -k);
seq(print(seq(simplify(T(n, k)), k=0..9)), n=0..6); # Peter Luschny, May 23 2014
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MATHEMATICA
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T[0, _] = T[_, 0] = 1;
T[n_, k_] := Sum[Binomial[n+j, 2j] k^j CatalanNumber[j], {j, 0, n}];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A336574
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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j) * binomial(k*n+j+1,n)/(k*n+j+1).
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+10
8
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1, 1, 3, 1, 3, 6, 1, 3, 15, 12, 1, 3, 24, 93, 24, 1, 3, 33, 255, 645, 48, 1, 3, 42, 498, 3102, 4791, 96, 1, 3, 51, 822, 8691, 40854, 37275, 192, 1, 3, 60, 1227, 18708, 164937, 566934, 299865, 384, 1, 3, 69, 1713, 34449, 464115, 3305868, 8164263, 2474025, 768
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k * (1 + 2 * A_k(x)).
T(n,k) = (1/(k*n+1)) * Sum_{j=0..n} 2^(n-j) * binomial(k*n+1,j) * binomial((k+1)*n-j,n-j).
T(n,k) = (1/n) * Sum_{j=0..n-1} (-1)^j * 3^(n-j) * binomial(n,j) * binomial((k+1)*n-j,n-1-j) for n > 0.
T(n,k) = (1/n) * Sum_{j=1..n} 3^j * 2^(n-j) * binomial(n,j) * binomial(k*n,j-1) for n > 0. (End)
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
3, 3, 3, 3, 3, 3, ...
6, 15, 24, 33, 42, 51, ...
12, 93, 255, 498, 822, 1227, ...
24, 645, 3102, 8691, 18708, 34449, ...
48, 4791, 40854, 164937, 464115, 1055838, ...
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MATHEMATICA
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T[n_, k_] := Sum[2^j * Binomial[n, j] * Binomial[k*n + j + 1, n]/(k*n + j + 1), {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 27 2020 *)
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PROG
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(PARI) T(n, k) = sum(j=0, n, 2^j*binomial(n, j)*binomial(k*n+j+1, n)/(k*n+j+1));
(PARI) T(n, k) = my(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k*(1+2*A)); polcoeff(A, n);
(PARI) T(n, k) = sum(j=0, n, 2^(n-j)*binomial(k*n+1, j)*binomial((k+1)*n-j, n-j))/(k*n+1);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A133305
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a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*4^i*5^(n-i), a(0) = 1.
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+10
7
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1, 5, 45, 505, 6345, 85405, 1204245, 17558705, 262577745, 4005148405, 62070886845, 974612606505, 15471084667545, 247876665109005, 4003225107031845, 65101209768055905, 1065128963164067745, 17520376884067071205, 289572455530026439245, 4806489064223483202905
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OFFSET
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0,2
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COMMENTS
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The Hankel transform of this sequence is 20^C(n+1,2). - Philippe Deléham, Oct 28 2007
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LINKS
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FORMULA
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G.f.: (1-z-sqrt(z^2-18*z+1))/(8*z).
a(n) = Sum_{k=0..n} A088617(n,k)*4^k.
a(n) = Sum_{k=0..n} A060693(n,k)*4^(n-k).
a(n) = Sum_{k=0..n} C(n+k, 2k)*4^k*C(k), C(n) given by A000108.
a(0) = 1, a(n) = a(n-1) + 4*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
Conjecture: (n+1)*a(n) + 9*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014
G.f.: 1/(1 - 5*x/(1 - 4*x/(1 - 5*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017
a(n) = hypergeom([-n, n + 1], [2], -4]). - Peter Luschny, Jan 08 2018
a(n) ~ 5^(1/4) * phi^(6*n + 3) / (2^(5/2) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Nov 21 2021
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MATHEMATICA
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a[n_] := Hypergeometric2F1[-n, n + 1, 2, -4];
CoefficientList[Series[(1-x-Sqrt[x^2-18*x+1])/(8*x), {x, 0, 50}], x] (* G. C. Greubel, Feb 10 2018 *)
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PROG
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(PARI) x='x+O('x^30); Vec((1-x-sqrt(x^2-18*x+1))/(8*x)) \\ G. C. Greubel, Feb 10 2018
(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-18*x+1))/(8*x))) // G. C. Greubel, Feb 10 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A349253
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G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - 2 * x * A(x)^2)).
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+10
6
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1, 3, 19, 169, 1753, 19795, 236035, 2923857, 37256881, 485202307, 6429346899, 86405569657, 1174917167881, 16134949855251, 223460304878467, 3117521211476641, 43771643214792033, 618045740600046211, 8770377489446217235, 125013010654218317385, 1789104455068153153849
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = 1 + 2 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} binomial(n+k,n-k) * 2^k * binomial(3*k,k) / (2*k+1).
a(n) = hypergeom([1/3, 2/3, -n, n + 1], [1/2, 1, 3/2], -(3/2)^3). - Peter Luschny, Nov 12 2021
a(n) ~ sqrt(315 + 31*sqrt(105)) * (31 + 3*sqrt(105))^n / (9 * sqrt(Pi) * 2^(2*n + 5/2) * n^(3/2)). - Vaclav Kotesovec, Nov 13 2021
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MATHEMATICA
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nmax = 20; A[_] = 0; Do[A[x_] = 1/((1 - x) (1 - 2 x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + 2 Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n + k, n - k] 2^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 20}]
a[n_] := HypergeometricPFQ[{1/3, 2/3, -n, n + 1}, {1/2, 1, 3/2}, -(3/2)^3];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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