A389157 Expansion of (1/x) * Series_Reversion( x / (1 + x + x^2 * (1 + x)^4) ).

1, 1, 2, 8, 31, 115, 450, 1856, 7827, 33463, 145191, 638375, 2836291, 12710855, 57396860, 260913698, 1193002715, 5483097347, 25316894055, 117379360711, 546254213859, 2550754973499, 11947667117356, 56120836828390, 264296782866443, 1247665340523211

Offset

0,3

Comments

Binomial transform of A389155.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Robert Israel, Linear recurrence of order 16

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A389155(k).

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+1,k) * binomial(n+3*k+1,n-2*k).

a(n) = (1/(n+1)) * [x^n] (1 + x + x^2 * (1 + x)^4)^(n+1).

D-finite with recurrence of order 16 (see link). - Robert Israel, Apr 07 2026

Mathematica

Table[(1/(n+1)) Coefficient[(1+x+x^2*(1+x)^4)^(n+1), x, n], {n, 0, 19}] (* Vincenzo Librandi, Sep 28 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^2*(1+x)^4))/x)
(Magma) R<x> := PolynomialRing(Rationals()); [ (1/(n+1))*Coefficient(((1 + x+ x^2 * (1 + x)^4))^(n+1), n) : n in [0..30] ]; // Vincenzo Librandi, Sep 28 2025

Crossrefs

Cf. A389155.

Sequence in context: A062456 A289610 A077838 * A216318 A018916 A391222

Adjacent sequences: A389154 A389155 A389156 * A389158 A389159 A389160

Keyword

nonn

Author

Seiichi Manyama, Sep 25 2025

Status

approved