A369267 - OEIS

%I #17 Feb 15 2024 04:21:25

%S 1,2,7,32,163,884,5009,29310,175750,1074264,6668825,41929970,

%T 266464579,1708829584,11044663663,71871779008,470495357634,

%U 3096311833496,20472771422946,135937759368388,906056228361095,6059922934991008,40657629626645463

%N Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^3)^2 ).

%H P. Bala, <a href="/A251592/a251592.pdf">Fractional iteration of a series inversion operator</a>

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

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+2,k) * binomial(3*n-3*k+1,n-3*k).

%F a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^2 * (1+x^3)^2 )^(n+1). - _Seiichi Manyama_, Feb 14 2024

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2/(1+x^3)^2)/x)

%o (PARI) a(n, s=3, t=2, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

%Y Cf. A369265, A369269.

%Y Cf. A369266, A370249.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 18 2024