A389659 Expansion of (1/x) * Series_Reversion( x / (1 + x^2 / (1 - x)^3)^2 ).

1, 0, 2, 6, 21, 86, 355, 1512, 6618, 29484, 133356, 610852, 2827567, 13206180, 62157680, 294530032, 1403871456, 6726584208, 32380629079, 156527780712, 759513525710, 3697982365088, 18061220726289, 88464072539580, 434433290576760, 2138584002325020, 10551091982565515

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

a(n) = (1/(n+1)) * [x^n] (1 + x^2 / (1 - x)^3)^(2*(n+1)).

Mathematica

Table[SeriesCoefficient[(1+x^2/(1-x)^3)^(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)^3)^2)/x)
(Magma) [1/(n+1)*&+[Binomial(2*n+2, k)*Binomial(n+k-1, n-2*k): k in [0..Floor(n/2)]]: n in [0..35]]; // Vincenzo Librandi, Oct 20 2025

Crossrefs

Cf. A187430, A389658.

Sequence in context: A344229 A090805 A150226 * A326335 A256180 A150227

Adjacent sequences: A389656 A389657 A389658 * A389660 A389661 A389662

Keyword

nonn

Author

Seiichi Manyama, Oct 10 2025

Status

approved