A389692 Expansion of (1/x) * Series_Reversion( x * (1 - x / (1 - x)^3)^2 ).

1, 2, 13, 96, 790, 6946, 63888, 607160, 5915019, 58756722, 592876446, 6059963002, 62612848686, 652888656558, 6861824438974, 72613429307862, 773046686960617, 8273778904753402, 88973164378574934, 960855080653506802, 10416427333293360147, 113314541082159748812, 1236583302947106903656

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Formula

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

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

G.f.: B(x)^2, where B(x) is the g.f. of A367241.

Mathematica

Table[(1/(n+1))*Sum[Binomial[2*n+k+1, k]* Binomial[n+2*k-1, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Oct 31 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x/(1-x)^3)^2)/x)
(Magma) [(1/(n+1))*&+[Binomial(2*n+k+1, k) * Binomial(n+2*k-1, n-k): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Oct 31 2025

Crossrefs

Cf. A211789, A389691.

Cf. A365150, A367241.

Sequence in context: A369202 A391464 A118352 * A372891 A320360 A074614

Adjacent sequences: A389689 A389690 A389691 * A389693 A389694 A389695

Keyword

nonn

Author

Seiichi Manyama, Oct 11 2025

Status

approved