%I #21 Mar 04 2024 14:06:04

%S 1,3,12,58,314,1824,11107,69955,451918,2977834,19936332,135225006,

%T 927267595,6417580459,44770275705,314489676679,2222549047262,

%U 15791353483602,112734135824404,808247711066688,5817056710700424,42012120642574732,304384379305912686

%N Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^2 + x) ).

%H Seiichi Manyama, <a href="/A369616/b369616.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

%F a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(3*n-3*k+1,n-k).

%F D-finite with recurrence 2*(n+1)*(2*n+1)*a(n) +3*(-13*n^2+1)*a(n-1) +33*(2*n-1)*(n-1)*a(n-2) -31*(n-1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Jan 28 2024

%p A369616 := proc(n)

%p add(binomial(n+1,k) * binomial(3*n-3*k+1,n-k),k=0..n) ;

%p %/(n+1) ;

%p end proc;

%p seq(A369616(n),n=0..70) ; # _R. J. Mathar_, Jan 28 2024

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1/(1-x)^2+x))/x)

%o (PARI) a(n) = sum(k=0, n, binomial(n+1, k)*binomial(3*n-3*k+1, n-k))/(n+1);

%Y Cf. A007317, A369617, A370844.

%Y Cf. A006013, A274734.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 27 2024