A389660 - OEIS

%I #14 Oct 20 2025 02:51:05

%S 1,2,11,68,473,3510,27251,218586,1797267,15067316,128307685,

%T 1106775150,9650428107,84920021864,753171175532,6725897514182,

%U 60424766404011,545740764344752,4952362528273121,45131545826450266,412865243836628363,3790014882563926310,34901419463705667915

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

%H Vincenzo Librandi, <a href="/A389660/b389660.txt">Table of n, a(n) for n = 0..1000</a>

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

%F a(n) = (1/(n+1)) * [x^n] 1/(1 - x * (1 + x)^2)^(2*(n+1)).

%t Table[SeriesCoefficient[1/(1-x*(1+x)^2)^(2*(n+1)),{x,0,n}]/(n+1),{n,0,30}] (* _Vincenzo Librandi_, Oct 20 2025 *)

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

%o (Magma) [1/(n+1)*&+[Binomial(2*n+k+1, k)*Binomial(2*k, n-k): k in [0..n]]: n in [0..35]]; // _Vincenzo Librandi_, Oct 20 2025

%Y Cf. A368961.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Oct 10 2025