%I #17 Oct 20 2025 01:39:11

%S 1,0,2,4,15,52,192,732,2853,11340,45756,186964,772057,3216912,

%T 13507936,57103516,242831553,1038056904,4458248878,19227639168,

%U 83239247342,361591141112,1575659134512,6885666506004,30169456622015,132506539685052,583278733961600,2572843291435872

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

%H Vincenzo Librandi, <a href="/A389658/b389658.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%t Table[SeriesCoefficient[(1+x^2/(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^2/(1-x)^2)^2)/x)

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

%Y Cf. A187430, A389659.

%Y Cf. A389245.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 10 2025