Displaying 1-10 of 22 results found.
1, 2, 11, 66, 461, 3448, 27061, 219702, 1829851, 15547142, 134224361, 1174119120, 10383783641, 92691197962, 834047700091, 7557110252538, 68890745834341, 631392034936040, 5814520777199261, 53776065007163886, 499275423496447211
FORMULA
G.f.: (1+x-x^2-sqrt(1-10*x-x^2+10*x^3+x^4))/(6*x*(1-x^2)).
G.f.: 1/(1-2x-x^2-6x^2/(1-5x-x^2-6x^2/(1-5x-x^2-6x^2/(1-5-x^2-6x^2/(1-... (continued fraction).
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)* A107841(n-2*k). Recurrence: (n+1)*a(n) = (n-5)*a(n-6) + 5*(2*n-7)*a(n-5) - (2*n-7)*a(n-4) - 20*(n-2)*a(n-3) + (2*n-1)*a(n-2) + 5*(2*n-1)*a(n-1). - R. J. Mathar, Jul 24 2012, simplified by Fung Lam, Jan 27 2014 a(n) ~ r*(r+10) * sqrt(10*r^3-2*r^2-30*r+4) / (12 * sqrt(Pi) * n^(3/2) * r^(n+1)), where r = 1 / (5/2 + sqrt(6) + 1/2*sqrt(53+20*sqrt(6))) = 0.100010105114224353... - Vaclav Kotesovec, Feb 27 2014
MATHEMATICA
CoefficientList[Series[(1+x-x^2-Sqrt[1-10x-x^2+10x^3+x^4])/(6x(1-x^2)), {x, 0, 20}], x] (* Harvey P. Dale, Aug 12 2011 *)
PROG
(PARI) x='x+O('x^30); Vec((1+x-x^2-sqrt(1-10*x-x^2+10*x^3+x^4))/(6*x*(1-x^2))) \\ G. C. Greubel, Apr 30 2018 (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1+x-x^2-Sqrt(1-10*x-x^2+10*x^3+x^4))/(6*x*(1-x^2)))) // G. C. Greubel, Apr 30 2018
1, 4, 24, 164, 1208, 9348, 74920, 616420, 5176296, 44182916, 382205048, 3343343268, 29523386968, 262826367748, 2356256046216, 21254326842596, 192766180154120, 1756758963727620, 16079466335134168, 147748236828875428, 1362397741935948024, 12603116216808465284, 116929440001191010664
FORMULA
Recurrence: (n+2)*a(n) = (4-n)*a(n-4) + 4*(2*n-5)*a(n-3) + 18*(n-1)*a(n-2) + 4*(2*n+1)*a(n-1), n>=4.
Recurrence (of order 2): (n+2)*(2*n-1)*a(n) = 4*(5*n^2-2)*a(n-1) - (n-2)*(2*n+1)*a(n-2). - Vaclav Kotesovec, Feb 27 2014 a(n) ~ sqrt(360+147*sqrt(6)) * (5+2*sqrt(6))^n / (9 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 27 2014
MATHEMATICA
CoefficientList[Series[((1 + x - Sqrt[1 - 10*x + x^2])/(6*x))^2, {x, 0, 100}], x] (* Vaclav Kotesovec, Feb 27 2014 *)
1, 8, 64, 520, 4304, 36232, 309504, 2677128, 23405520, 206522888, 1836913216, 16452907016, 148274884688, 1343569891720, 12233903203328, 111883174439304, 1027244073375312, 9465236716896264, 87498251217286720, 811252609543727624, 7542152541765899728, 70294794046928531848
FORMULA
Recurrence: (n+4)*a(n) = (8-n)*a(n-8) + 4*(4*n-26)*a(n-7) + 64*(5-n)*a(n-6) + 8*(2*n-7)*a(n-5) + 194*(n-2)*a(n-4) + 8*(2*n-1)*a(n-3) - 64*(n+1)*a(n-2) + 8*(2*n+5)*a(n-1), n>=8.
Recurrence (of order 2): n*(n+4)*(2*n+1)*a(n) = 20*n*(n+1)*(n+2)*a(n-1) - (n-2)*(n+2)*(2*n+3)*a(n-2). - Vaclav Kotesovec, Feb 27 2014 a(n) ~ 2*sqrt(35280+14403*sqrt(6)) * (5+2*sqrt(6))^n / (27 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 27 2014
MATHEMATICA
CoefficientList[Series[((1+x-Sqrt[1-10*x+x^2])/(6*x))^4, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 27 2014 *)
1, 2, 9, 58, 401, 2952, 22759, 181358, 1481751, 12346102, 104505959, 896170608, 7768885801, 67972510202, 599449125609, 5323095489058, 47555513297801, 427127946025752, 3854618439044959, 34934658168463958, 317834095671077751, 2901725605879035502, 26575914921615695759
COMMENTS
This is the Chebyshev transform over the positive strip 0<=x<=1. A160852 may be viewed as the Chebyshev transform over the negative strip -1<=x<=0.
FORMULA
G.f.: (1+x+x^2 - sqrt(1-10*x+3*x^2-10*x^3+x^4))/(6*x*(1+x^2)).
G.f.: F(x/(1+x^2)), where F(x) is the g.f. of A107841. Recurrence: (n+1)*a(n) = (5-n)*a(n-6) + 5*(2*n-7)*a(n-5) + (11-4*n)*a(n-4)
+ 20*(n-2)*a(n-3) + (5-4*n)*a(n-2) + 5*(2*n-1)*a(n-1), n>=6.
a(n) ~ (sqrt(45+20*sqrt(6))/2+sqrt(6)+5/2)^n*sqrt(120-30*sqrt(6)+2*sqrt(30*(6196*sqrt(6)-15159)))/(12*sqrt(Pi*n^3)).
MATHEMATICA
CoefficientList[Series[(1+x+x^2 - Sqrt[1-10*x+3*x^2-10*x^3+x^4])/(6*x*(1+x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 30 2014 *)
PROG
(PARI) x='x+O('x^50); Vec((1+x+x^2 - sqrt(1-10*x+3*x^2-10*x^3+x^4))/(6*x*(1+x^2))) \\ G. C. Greubel, Apr 05 2017
Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.
+10
38
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
COMMENTS
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
REFERENCES
Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 449.
FORMULA
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
EXAMPLE
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;
MAPLE
R := n -> simplify(hypergeom([-n, n + 1], [2], -x)):
Trow := n -> seq(coeff(R(n, x), x, k), k = 0..n):
MATHEMATICA
Table[Binomial[n+k, n] Binomial[n, k]/(k+1), {n, 0, 10}, {k, 0, n}]//Flatten (* Michael De Vlieger, Aug 10 2017 *)
PROG
(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
Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282.
+10
26
1, 0, 1, 0, 1, 2, 0, 1, 5, 5, 0, 1, 9, 21, 14, 0, 1, 14, 56, 84, 42, 0, 1, 20, 120, 300, 330, 132, 0, 1, 27, 225, 825, 1485, 1287, 429, 0, 1, 35, 385, 1925, 5005, 7007, 5005, 1430, 0, 1, 44, 616, 4004, 14014, 28028, 32032, 19448, 4862, 0, 1, 54, 936, 7644, 34398, 91728
COMMENTS
With polynomials
P(0,t) = 0
P(1,t) = 1
P(2,t) = t
P(3,t) = t + 2 t^2
P(4,t) = t + 5 t^2 + 5 t^3
P(5,t) = t + 9 t^2 + 21 t^3 + 14 t^4
The o.g.f. A(x,t) = {1+x-sqrt[(1-x)^2-4xt]}/[2(1+t)] (see Drake et al.).
B(x,t)= x-t x^2/(1-x)= x-t(x^2+x^3+x^4+...) is the comp. inverse in x.
Let h(x,t) = 1/(dB/dx) = (1-x)^2/(1+(1+t)*x*(x-2)) = 1/(1-t(2x+3x^2+4x^3+...)), an o.g.f. in x for row polynomials in t of A181289. Then P(n,t) is given by (1/n!)(h(x,t)*d/dx)^n x, evaluated at x=0, A = exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t). These results are a special case of A133437 with u(x,t) = B(x,t), i.e., u_1=1 and (u_n)=-t for n > 1. See A001003 for t = 1. (End) Let U(x,t) = [A(x,t)-x]/t, then U(x,0) = -dB(x,t)/dt and U satisfies dU/dt = UdU/dx, the inviscid Burgers' equation (Wikipedia), also called the Hopf equation (see Buchstaber et al.). Also U(x,t) = U(A(x,t),0) = U(x+tU,0) since U(x,0) = [x-B(x,t)]/t. - Tom Copeland, Mar 12 2012 T(r, s) is the number of [0,r]-covering hierarchies with s segments (see Kreweras). - Michel Marcus, Nov 22 2014 T(n,k) is the number of small Schröder n-paths (lattice paths from (0,0) to (2n,0) using steps U=(1,1), F=(2,0), D=(1,-1) with no F step on the x-axis) that has exactly k U steps.
T(n,k) is the number of Schröder trees (plane rooted tree where each internal node has at least two children) with exactly n+1 leaves and k internal nodes. (End)
LINKS
G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973, p. 21-22. G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
FORMULA
Triangle T(n, k) read by rows; given by [0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938. For k>0, T(n, k) = binomial(n-1, k-1)*binomial(n+k, k)/(n+1); T(0, 0) = 1 and T(n, 0) = 0 if n > 0. [corrected by Marko Riedel, May 04 2023] Sum_{k=0..n} T(n,k)*x^k = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Nov 05 2007 Umbrally, P(n,t) = Lah[n-1,-t*a.]/n! = (1/n)*Sum_{k=1..n-1} binomial(n-2,k-1)a_k t^k/k!, where (a.)^k = a_k = (n-1+k)!/(n-1)!, the rising factorial, and Lah(n,t) = n!*Laguerre(n,-1,t) are the Lah polynomials A008297 related to the Laguerre polynomials of order -1. - Tom Copeland, Oct 04 2014 T(n, k) = binomial(n, k)*binomial(n+k, k-1)/n, for k >= 0; T(0, 0) = 1 (see Kreweras, p. 21). - Michel Marcus, Nov 22 2014 P(n,t) = Lah[n-1,-:Dt:]/n! t^(n-1) with (:Dt:)^k = (d/dt)^k t^k = k! Laguerre(k,0,-:tD:) with (:tD:)^j = t^j D^j. The normalized Laguerre polynomials of 0 order are given in A021009. - Tom Copeland, Aug 22 2016
EXAMPLE
Triangle starts:
1;
0, 1;
0, 1, 2;
0, 1, 5, 5;
0, 1, 9, 21, 14;
...
MATHEMATICA
Table[Boole[n == 2] + If[# == -1, 0, Binomial[n - 3, #] Binomial[n + # - 1, #]/(# + 1)] &[k - 1], {n, 2, 12}, {k, 0, n - 2}] // Flatten (* after Jean-François Alcover at A033282, or *) Table[If[n == 0, 1, Binomial[n, k] Binomial[n + k, k - 1]/n], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 22 2016 *)
PROG
(PARI) t(n, k) = if (n==0, 1, binomial(n, k)*binomial(n+k, k-1)/n);
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Nov 22 2014
CROSSREFS
Diagonals: A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279, A000108, A002054, A002055, A002056, A007160, A033280, A033281. Row sums: A001003 (Schroeder numbers).
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.
+10
25
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
COMMENTS
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
FORMULA
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
EXAMPLE
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]
...
MATHEMATICA
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 *)
PROG
(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
a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*2^i*3^(n-i), a(0)=1.
+10
24
1, 3, 15, 93, 645, 4791, 37275, 299865, 2474025, 20819307, 178003815, 1541918901, 13503125805, 119352115551, 1063366539315, 9539785668657, 86104685123025, 781343125570515, 7124072211203775, 65233526296899981, 599633539433039445, 5531156299278726663
COMMENTS
The Hankel transform of this sequence is 6^C(n+1,2). - Philippe Deléham, Oct 28 2007 The Hankel transform of the sequence starting 1, 1, 3, 15, ... is A081955. - Paul Barry, Dec 09 2008 Number of Schroeder paths from (0,0) to (0,2n) allowing two colors for the down steps (or alternatively for the rise steps). - Paul Barry, Feb 01 2009 Essentially, reversion of x*(1-2*x)/(1+x). - Paul Barry, Apr 28 2009 a(n) is also the number of infix expressions with n variables and operators +, - and * (or +, * and /) such that there are no redundant parentheses. - Vjeran Crnjak, Apr 25 2020
FORMULA
G.f.: (1 - z - sqrt(1 -10*z +z^2))/(4*z).
a(n) = Sum_{k=0..n} C(n+k, 2k) * 2^k * C(k), C(n) given by A000108. - Paul Barry, May 21 2005 a(0) = 1, a(n) = a(n-1) + 2*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007 G.f.: 1/(1-x-2*x/(1-x-2*x/(1-x-2*x/(1-.... (continued fraction). - Paul Barry, Feb 01 2009 G.f.: 1/(1-3*x-6*x^2/(1-5*x-6*x^2/(1-5*x-6*x^2/(1-... (continued fraction). - Paul Barry, Apr 28 2009 G.f.: 1/(1-3*x/(1-2*x/(1-3*x/(1-2*x/(1-3*x/(1-... (continued fraction). - Paul Barry, May 14 2009 a(n) = Hypergeometric2F1(-n,n+1;2;-2) = Sum_{k=0..n} C(n+k,k) * C(n,k) * 2^k/(k+1). - Paul Barry, Feb 08 2011 G.f.: A(x) = (1-x-(x^2-10*x+1)^(1/2))/(4*x) = 1/(G(0)-x); G(k)= 1 + x - 3*x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 05 2012 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/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012 G.f. A(x) satisfies: A(x) = (1 + 2*x*A(x)^2) / (1 - x). - Ilya Gutkovskiy, Jun 30 2020
MAPLE
if n = 0 then
1;
else
add(binomial(n, i)*binomial(n, i+1)*2^i*3^(n-i), i=0..n-1)/n ;
end if;
A103210_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 3*a[w-1] + 2*add(a[j]*a[w-j-1], j=1..w-1) od;
MATHEMATICA
CoefficientList[Series[(1-x-Sqrt[x^2-10*x+1])/(4*x), {x, 0, 25}], x] (* Vaclav Kotesovec, Oct 17 2012 *)
PROG
(PARI) my(x='x+O('x^25)); Vec((1-x-sqrt(x^2-10*x+1))/(4*x)) \\ G. C. Greubel, Feb 10 2018 (Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!((1-x-Sqrt(x^2-10*x+1))/(4*x))); // G. C. Greubel, Feb 10 2018 (Sage) [1]+[(3^n/n)*sum( binomial(n, j)*binomial(n, j+1)*(2/3)^j for j in (0..n-1)) for n in (1..25)] # G. C. Greubel, Jun 08 2020
Series reversion of x*(1-3*x^2)/(1-x^2) in odd-order powers.
+10
8
1, 2, 14, 130, 1382, 15906, 192894, 2427522, 31405430, 415086658, 5580629870, 76080887042, 1049295082630, 14613980359010, 205246677882078, 2903566870820610, 41337029956899222, 591796707042765954, 8514525059135909070, 123048063153362454402
COMMENTS
This sequence is implied in the solutions of magnetohydrodynamics equations in R^3 for incompressible, electrically-conducting fluids in the presence of a strong Lorentz force. a(n) = numbers of allowable magneto-vortical eddies in terms of initial conditions.
FORMULA
G.f.: (exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v + x/9)/x, where i=sqrt(-1),
u = (1/9)*(x^3 - 108 *x + 9*sqrt(-9 + 141*x^2 - 3*x^4))^(1/3), and
v = (1/9)*(x^3 - 108 *x - 9*sqrt(-9 + 141*x^2 - 3*x^4))^(1/3).
a(n) = [x^n] 2*Sum_{j = 1..n} ((Sum_{k = 1..n} a(k)*x^(2*k-1))^(2*j+1)), a(1) = 1, with offset by 1.
D-finite with recurrence 12*n*(2*n+1)*a(n) +(-382*n^2+391*n-90)*a(n-1) +3*(34*n^2-132*n+125)*a(n-2) -(2*n-5)*(n-3)*a(n-3)=0. - R. J. Mathar, Mar 24 2023 a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(2*n+k+1,n) / (2*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * 3^(n-k) * binomial(n,k) * binomial(2*n,k-1) for n > 0. (End)
a(n) = 2*Jacobi_P(n-1, 1, n+1, 5)/n for n >= 1.
Second-order recurrence: 3*n*(2*n + 1)*(13*n - 17)*a(n) = (1222*n^3 - 2820*n^2 + 1877*n - 360)*a(n-1) - (n - 2)*(13*n - 4)*(2*n - 3)*a(n-2) with a(0) = 1 and a(1) = 2. (End)
MAPLE
Order := 60 ;
solve(series(x*(1-3*x^2)/(1-x^2), x)=y, x) ;
convert(%, polynom) ;
seq(coeff(%, y, 2*i+1), i=0..Order/2) ; # R. J. Mathar, Jul 20 2023
MATHEMATICA
Table[(CoefficientList[InverseSeries[Series[x*(1-3*x^2)/(1-x^2), {x, 0, 40}], x], x])[[n]], {n, 2, 40, 2}] (* Vaclav Kotesovec, Jan 29 2014 *)
PROG
(PARI) v=Vec( serreverse(x*(1-3*x^2)/(1-x^2) +O(x^66) ) ); vector(#v\2, j, v[2*j-1]) \\ Joerg Arndt, Jan 14 2014
Series reversion of x*(1-4*x)/(1-x) is x*A(x) where A(x) is the generating function.
+10
7
1, 3, 21, 183, 1785, 18651, 204141, 2310447, 26819121, 317530227, 3819724293, 46553474919, 573608632233, 7133530172619, 89423593269213, 1128765846337887, 14334721079385441, 183021615646831587, 2347944226115977461, 30250309354902101271, 391241497991342192985
COMMENTS
The Hankel transform of this sequence is 12^C(n+1,2).
Number of Dyck n-paths with two colors of up (U,u) and two colors of down (D,d) avoiding UD. - David Scambler, Jun 24 2013 Number of small Schröder n-paths with 3 types of up steps (i.e., lattice paths from (0,0) to (2n,0) using steps U1=U2=U3=(1,1), F=(2,0), D=(1,-1), with no F steps on the x-axis). - Yu Hin Au, Dec 05 2019
FORMULA
a(n) = Sum_{0<=k<=n} A086810(n,k)*3^k. D-finite with recurrence: (n+1)*a(n) +7*(-2*n+1)*a(n-1) +(n-2)*a(n-2)=0. - R. J. Mathar, Aug 16 2015 a(n) = (-1)^n*hypergeom([-n, n + 1], [2], 4). - Peter Luschny, Jan 08 2018 G.f.: (1 + x - sqrt(1 - 14*x + x^2))/(8*x). - Michael Somos, Jul 27 2022 Given g.f. A(x) and y = 2*x*A(-x^2), then y-1/y = (x-1/x)/2.
If a(n) := -a(-1-n) for n<0, then 0 = a(n)*(+a(n+1) -35*a(n+2) +4*a(n+3)) +a(n+1)*(+7*a(n+1) +194*a(n+2) -35*a(n+3)) +a(n+2)*(+7*a(n+2) +a(n+3)) for all n in Z. (End)
EXAMPLE
G.f. = 1 + 3*x + 21*x^2 + 183*x^3 + 1785*x^4 + 18651*x^5 + ... - Michael Somos, Jul 27 2022
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[x*(1-4*x)/(1-x), {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Mar 30 2015 *) Table[(-1)^n Hypergeometric2F1[-n, n + 1, 2, 4], {n, 0, 20}] (* Peter Luschny, Jan 08 2018 *) a[ n_] := SeriesCoefficient[(1 + x - Sqrt[1 - 14*x + x^2])/(8*x), {x, 0, n}]; (* Michael Somos, Jul 27 2022 *) a[ n_] := (-1)^n * Hypergeometric2F1[ -n, n+1, 2, 4]; (* Michael Somos, Mar 15 2024 *)
PROG
(PARI) Vec(serreverse(x*(1-4*x)/(1-x)+ O(x^30))) \\ Michel Marcus, Mar 30 2015 (PARI) {a(n) = if(n<0, 0, n++; polcoeff(serreverse(x*(1-4*x)/(1-x) + x*O(x^n)), n))}; /* Michael Somos, Jul 27 2022 */ (PARI) {a(n) = if(n<0, -a(-1-n), polcoeff(2/(1 + x + sqrt(1 - 14*x + x^2 + x*O(x^n))), n))}; /* Michael Somos, Mar 15 2024 */
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