A389158 Expansion of (1/x) * Series_Reversion( x / (1 + x + x^3 * (1 + x)^4) ).

1, 1, 1, 2, 9, 37, 124, 380, 1213, 4249, 15633, 57410, 208168, 755418, 2777484, 10352457, 38878837, 146454253, 553090105, 2096803160, 7984862557, 30528384971, 117081373080, 450181102730, 1735173140972, 6704158816736, 25962259079446, 100751154896056, 391724555617936

Offset

0,4

Comments

Binomial transform of A389156.

Links

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

Robert Israel, Linear recurrence of order 20

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A389156(k).

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

a(n) = (1/(n+1)) * [x^n] (1 + x + x^3 * (1 + x)^4)^(n+1).

D-finite with recurrence of order 20 (see link). - Robert Israel, Apr 07 2026

Mathematica

Table[(1/(n+1)) Coefficient[(1+x+x^3*(1+x)^4)^(n+1), x, n], {n, 0, 29}] (* Vincenzo Librandi, Sep 28 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^3*(1+x)^4))/x)
(Magma) R<x> := PolynomialRing(Rationals()); [ (1/(n+1))*Coefficient(((1 + x+ x^3 * (1 + x)^4))^(n+1), n) : n in [0..30] ]; // Vincenzo Librandi, Sep 28 2025

Crossrefs

Cf. A389156.

Sequence in context: A389445 A373910 A373909 * A389441 A206374 A037553

Adjacent sequences: A389155 A389156 A389157 * A389159 A389160 A389161

Keyword

nonn

Author

Seiichi Manyama, Sep 25 2025

Status

approved