A389662 Expansion of (1/x) * Series_Reversion( x * (1 - x^2 * (1 + x)^2)^2 ).

1, 0, 2, 4, 13, 52, 170, 680, 2584, 10164, 40940, 165100, 678069, 2801284, 11670384, 48972308, 206622615, 876685992, 3736757466, 15995113684, 68729224795, 296334919160, 1281723760100, 5559690003140, 24179743368417, 105416266216908, 460614365084180, 2016836839302072

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

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

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

Mathematica

Table[SeriesCoefficient[1/(1-x^2*(1+x)^2)^(2*(n+1)), {x, 0, n}]/(n+1), {n, 0, 30}] (* Vincenzo Librandi, Oct 19 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x^2*(1+x)^2)^2)/x)
(Magma) [1/(n+1)*&+[Binomial(2*n+k+1, k)*Binomial(2*k, n-2*k): k in [0..Floor(n/2)]]: n in [0..35]]; // Vincenzo Librandi, Oct 19 2025

Crossrefs

Cf. A389294.

Sequence in context: A069730 A072605 A330344 * A161905 A030953 A030811

Adjacent sequences: A389659 A389660 A389661 * A389663 A389664 A389665

Keyword

nonn

Author

Seiichi Manyama, Oct 10 2025

Status

approved