A389063 Expansion of (1/x) * Series_Reversion( x / (1 + x^2 * (1 + x)^3) ).

1, 0, 1, 3, 5, 16, 50, 133, 410, 1287, 3917, 12375, 39600, 126490, 409110, 1333367, 4360470, 14346708, 47436086, 157405329, 524343950, 1752742875, 5876285021, 19756979990, 66599448577, 225031535515, 762033353850, 2585794709160, 8790962711325

Offset

0,4

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(3*k,n-2*k).

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

Mathematica

Table[(1/(n+1)) Coefficient[(1+x^2*(1+x)^3)^(n+1), x, n], {n, 0, 35}] (* Vincenzo Librandi, Sep 29 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x^2*(1+x)^3))/x)
(Magma) R<x> := PolynomialRing(Rationals()); [ (1/(n+1))*Coefficient(((1 + x^2 * (1 + x)^3)^(n+1)), n) : n in [0..30] ]; // Vincenzo Librandi, Sep 29 2025

Crossrefs

Cf. A378425.

Sequence in context: A243321 A099101 A208819 * A038120 A192911 A105408

Adjacent sequences: A389060 A389061 A389062 * A389064 A389065 A389066

Keyword

nonn

Author

Seiichi Manyama, Sep 22 2025

Status

approved