A153232 a(n) = Sum((-1)^(n-i)*binomial(3i,i)*binomial(n+2i,3i)*2^i/(2i+1),i=0..n).

1, 1, 5, 31, 217, 1637, 12985, 106767, 901857, 7779321, 68238029, 606829087, 5458425065, 49575244557, 454002681969, 4187624605215, 38868943003201, 362779053190001, 3402660186792277, 32055797319184735, 303189929508106169, 2877922441328068981, 27406770566397160361

Offset

0,3

Comments

a(n) is also the number of h-avoiding generalized noncrossing trees of n+1 points.

Links

Jon E. Schoenfield, Table of n, a(n) for n=0..100

Formula

Recurrence: n*(2*n+1)*a(n) = (25*n^2 - 31*n + 9)*a(n-1) - 3*(16*n^2 - 61*n + 57)*a(n-2) + 2*(n-3)*(5*n-14)*a(n-3) + 4*(n-4)*(n-3)*a(n-4). - Vaclav Kotesovec, Oct 24 2012

a(n) ~ sqrt(1+sqrt(3))*(5+3*sqrt(3))^n/(2*sqrt(6*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

a(n) = (-1)^n* 3F2(-n, 1+n/2, (n+1)/2; 1, 3/2; 2). - Vaclav Kotesovec, Oct 27 2012

G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(x)^2). - Ilya Gutkovskiy, Nov 03 2021

Mathematica

Table[Sum[(-1)^(n-k)*Binomial[3*k, k]*Binomial[n+2*k, 3*k]*2^k/(2*k+1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 24 2012 *)
Table[(-1)^n*HypergeometricPFQ[{(n+1)/2, 1+n/2, -n}, {1, 3/2}, 2], {n, 0, 20}] (* Vaclav Kotesovec, Oct 27 2012 *)

Crossrefs

Sequence in context: A146962 A269730 A036758 * A287899 A110379 A097146

Adjacent sequences: A153229 A153230 A153231 * A153233 A153234 A153235

Keyword

nonn

Author

Yidong Sun (sydmath(AT)yahoo.com.cn), Dec 21 2008

Extensions

Edited, corrected (offset) and extended by Jon E. Schoenfield, Dec 26 2008

Status

approved