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
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