%I #23 Nov 03 2021 12:44:39
%S 1,1,5,31,217,1637,12985,106767,901857,7779321,68238029,606829087,
%T 5458425065,49575244557,454002681969,4187624605215,38868943003201,
%U 362779053190001,3402660186792277,32055797319184735,303189929508106169,2877922441328068981,27406770566397160361
%N a(n) = Sum((-1)^(n-i)*binomial(3i,i)*binomial(n+2i,3i)*2^i/(2i+1),i=0..n).
%C a(n) is also the number of h-avoiding generalized noncrossing trees of n+1 points.
%H Jon E. Schoenfield, <a href="/A153232/b153232.txt">Table of n, a(n) for n=0..100</a>
%F 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
%F a(n) ~ sqrt(1+sqrt(3))*(5+3*sqrt(3))^n/(2*sqrt(6*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2012
%F a(n) = (-1)^n* 3F2(-n, 1+n/2, (n+1)/2; 1, 3/2; 2). - _Vaclav Kotesovec_, Oct 27 2012
%F G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(x)^2). - _Ilya Gutkovskiy_, Nov 03 2021
%t 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 *)
%t Table[(-1)^n*HypergeometricPFQ[{(n+1)/2,1+n/2,-n},{1,3/2},2],{n,0,20}] (* _Vaclav Kotesovec_, Oct 27 2012 *)
%K nonn
%O 0,3
%A Yidong Sun (sydmath(AT)yahoo.com.cn), Dec 21 2008
%E Edited, corrected (offset) and extended by _Jon E. Schoenfield_, Dec 26 2008
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