A388914 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (2*x+(1+x)^2)) ).

1, 6, 46, 402, 3801, 37866, 391484, 4161114, 45188596, 499211928, 5592563046, 63384728622, 725474517694, 8373535909212, 97354076497272, 1139095054916874, 13402911355404588, 158489091634248264, 1882485494028832856, 22449231615558037512, 268684367827092688937

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 2^k * binomial(n+1,k) * binomial(4*n-2*k+4,n-k).

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

Mathematica

Table[(1/(n+1)) Coefficient[((1+x)^2*(2*x+(1+x)^2))^(n+1), x, n], {n, 0, 20}] (* Vincenzo Librandi, Oct 01 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2*(2*x+(1+x)^2)))/x)
(Magma) R<x> := PolynomialRing(Rationals()); a := func< n | Coefficient(((1+x)^2 * (2*x + (1+x)^2))^(n+1), n)/(n+1)>; [a(n):n in [0..20]]; // Vincenzo Librandi, Oct 01 2025

Crossrefs

Cf. A118346, A364792.

Sequence in context: A380685 A390622 A084772 * A199563 A349332 A365185

Adjacent sequences: A388911 A388912 A388913 * A388915 A388916 A388917

Keyword

nonn

Author

Seiichi Manyama, Sep 21 2025

Status

approved