A389407 Expansion of (1/x) * Series_Reversion( x * (1 - x^3 * (1 + x)^3) ).

1, 0, 0, 1, 3, 3, 5, 27, 75, 132, 324, 1131, 3087, 7080, 19176, 58548, 161082, 417525, 1168699, 3410946, 9525450, 26124550, 74177610, 213755490, 605003256, 1705277691, 4882522041, 14056846708, 40196816748, 114951379620, 331136162940, 956168299020, 2755091804124, 7946845872290

Offset

0,5

Links

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

Formula

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

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

Mathematica

Table[SeriesCoefficient[1/(1-x^3*(1+x)^3)^(n+1), {x, 0, n}]/(n+1), {n, 0, 30}] (* Vincenzo Librandi, Oct 17 2025 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1-x^3*(1+x)^3))/x)
(Magma) [1/(n+1)*&+[Binomial(n+k, k)*Binomial(3*k, n-3*k): k in [0..Floor(n/3)]]: n in [0..35]]; // Vincenzo Librandi, Oct 17 2025

Crossrefs

Cf. A361305, A389295.

Sequence in context: A384891 A366594 A265878 * A201496 A110426 A200562

Adjacent sequences: A389404 A389405 A389406 * A389408 A389409 A389410

Keyword

nonn

Author

Seiichi Manyama, Oct 03 2025

Status

approved