A389132 Expansion of (1/x) * Series_Reversion( x / (1 + x + x^4 * (1 + x)^2) ).

1, 1, 1, 1, 2, 8, 29, 85, 215, 517, 1315, 3719, 11216, 33840, 99282, 284972, 817161, 2381021, 7070080, 21220132, 63781443, 191222879, 572686456, 1719262030, 5185559991, 15711884201, 47748474606, 145325700586, 442724886406, 1350355791694, 4125710889838, 12629468711408

Offset

0,5

Comments

Binomial transform of A389130.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A389130(k).

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

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

Mathematica

Table[(1/(n+1)) Coefficient[(1+x+x^4*(1+x)^2)^(n+1), x, n], {n, 0, 35}] (* Vincenzo Librandi, Sep 28 2025 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^4*(1+x)^2))/x)
(Magma) R<x> := PolynomialRing(Rationals()); [ (1/(n+1))*Coefficient(((1 + x + x^4 * (1 + x)^2)^(n+1)), n) : n in [0..30] ]; // Vincenzo Librandi, Sep 28 2025

Crossrefs

Cf. A389130.

Sequence in context: A365760 A373906 A365758 * A389444 A365759 A373905

Adjacent sequences: A389129 A389130 A389131 * A389133 A389134 A389135

Keyword

nonn

Author

Seiichi Manyama, Sep 24 2025

Status

approved