A387497 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x^2*(1+x)^2)) ).

1, 2, 6, 24, 110, 536, 2713, 14142, 75472, 410396, 2265624, 12664236, 71532966, 407658812, 2341087610, 13534387376, 78705975664, 460080104896, 2701914889715, 15933668505570, 94316894224440, 560197450733384, 3337638150339424, 19942023081439884, 119462176246924346, 717354981405663892

Offset

0,2

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(n+1,k) * binomial(2*n+2*k+2,n-2*k).

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

Mathematica

Table[SeriesCoefficient[((1+x)^2*(1+x^2*(1+x)^2))^(n+1), {x, 0, n}]/(n+1), {n, 0, 30}] (* Vincenzo Librandi, Oct 20 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2*(1+x^2*(1+x)^2)))/x)
(Magma) [1/(n+1)*&+[Binomial(n+1, k)*Binomial(2*n+2*k+2, n-2*k): k in [0..Floor(n/2)]]: n in [0..35]]; // Vincenzo Librandi, Oct 20 2025

Crossrefs

Cf. A369439, A387496, A387498.

Sequence in context: A177519 A214762 A141254 * A366706 A216879 A372527

Adjacent sequences: A387494 A387495 A387496 * A387498 A387499 A387500

Keyword

nonn

Author

Seiichi Manyama, Oct 06 2025

Status

approved