OFFSET
0,2
COMMENTS
a(n) is the number of grand Motzkin paths from (0,0) to (2n+2,0) that avoid vertices (k,0) for all odd k and end on a down step. - Alexander Burstein, May 11 2021
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
Veronika Irvine, Stephen Melczer, and Frank Ruskey, Vertically constrained Motzkin-like paths inspired by bobbin lace, arXiv:1804.08725 [math.CO], 2018.
FORMULA
a(n) = (2/Pi)*Integral_{x=-1..1} (1+2*x)^(2*n+1)*sqrt(1-x^2). [Peter Luschny, Sep 11 2011]
Recurrence: (n+1)*(2*n+3)*a(n) = (14*n^2+23*n+6)*a(n-1) + 3*(14*n^2-37*n+21)*a(n-2) - 27*(n-2)*(2*n-3)*a(n-3). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ 3^(2*n+5/2)/(4*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
G.f.: (1/x) * Series_Reversion( x*(1+x) / ( (1+2*x)^2 * (1+x+x^2) ) ). - Paul D. Hanna, Oct 03 2014
From Peter Bala, Apr 20 2024: (Start)
a(n) = Sum_{k = 0..2*n+1} (-1)^(k+1) * binomial(2*n+1, k)*Catalan(k+1).
a(n) = Sum_{k = 0..2*n+1} (-1)^k * binomial(2*n+1, k)*Catalan(k+1)*3^(2*n-k+1).
(4*n - 1)*(2*n + 3)*(n + 1)*a(n) = 2*(4*n + 1)*(10*n^2 + 5*n - 3)*a(n-1) - 9*(4*n + 3)*(2*n - 1)*(n - 1)*a(n-2) with a(0) = 1 and a(1) = 4. (End)
MAPLE
G:=(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2): GG:=series(G, x=0, 60): seq(coeff(GG, x^(2*n-1)), n=1..24); # Emeric Deutsch
M := proc(n) option remember; `if`(n<2, 1, (3*(n-1)*M(n-2)+(2*n+1)*M(n-1))/(n+2)) end: A099250 := n -> M(2*n+1):
seq(A099250(i), i=0..20); # Peter Luschny, Sep 11 2011
MATHEMATICA
Take[CoefficientList[Series[(1-x-(1-2x-3x^2)^(1/2))/(2x^2), {x, 0, 60}], x], {2, -1, 2}] (* Harvey P. Dale, Sep 11 2011 *)
Table[Hypergeometric2F1[-1/2-n, -n, 2, 4], {n, 0, 30}] (* Jean-François Alcover, Apr 03 2015 *)
MotzkinNumber = DifferenceRoot[Function[{y, n}, {(-3n-3)*y[n] + (-2n-5)*y[n+1] + (n+4)*y[n+2] == 0, y[0] == 1, y[1] == 1}]];
Table[MotzkinNumber[2n+1], {n, 0, 20}] (* Jean-François Alcover, Oct 27 2021 *)
PROG
(PARI) a(n)=sum(j=0, 3*n+4, binomial(j, 2*j-5*n-7)*binomial(3*n+4, j))/(3*n+4); /* Joerg Arndt, Mar 09 2013 */
(PARI) x='x+O('x^66); v=Vec((1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2)); vector(#v\2, n, v[2*n]) \\ Joerg Arndt, May 12 2013
(PARI) {a(n)=polcoeff(1/x*serreverse( x*(1+x)/((1+2*x)^2*(1+x+x^2) +x^2*O(x^n)) ), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 03 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 16 2004
EXTENSIONS
More terms from Emeric Deutsch, Nov 17 2004
STATUS
approved