A389250 Expansion of (1/x) * Series_Reversion( x / (1 + x^3 / (1 - x)^3) ).

1, 0, 0, 1, 3, 6, 13, 36, 105, 292, 801, 2256, 6512, 18942, 55224, 161889, 478089, 1420368, 4237930, 12692115, 38153577, 115097364, 348298083, 1056929742, 3215561184, 9806369976, 29972578065, 91797401605, 281684569491, 865898757738, 2666201539741, 8222328605748

Offset

0,5

Links

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

Formula

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

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

Mathematica

a[n_]:=SeriesCoefficient[(1+x^3/(1-x)^3)^(n+1), {x, 0, n}]/(n+1); Table[a[n], {n, 0, 25}] (* Vincenzo Librandi, Oct 03 2025 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(serreverse(x/(1+x^3/(1-x)^3))/x)
(Magma) [&+[Binomial(n+1, k) * Binomial(n-1, n-3*k) : k in [0..Floor(n/3)] ] div (n+1): n in [0..30]]; // Vincenzo Librandi, Oct 03 2025

Crossrefs

Cf. A161797, A389246.

Sequence in context: A391831 A366940 A053564 * A264236 A216999 A036781

Adjacent sequences: A389247 A389248 A389249 * A389251 A389252 A389253

Keyword

nonn

Author

Seiichi Manyama, Sep 26 2025

Status

approved