A387666 Expansion of (1/x) * Series_Reversion( x * (1 - x * (1 + x)^3) / (1 + x)^2 ).

1, 3, 16, 109, 835, 6859, 59045, 525707, 4801180, 44728430, 423401822, 4060824494, 39376169016, 385379656475, 3801993127383, 37769799739738, 377500099243942, 3793338676749017, 38300291165225130, 388366933387600778, 3953284366725588494, 40382340111378123026

Offset

0,2

Links

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

Formula

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

a(n) = (1/(n+1)) * [x^n] ((1 + x)^2 / (1 - x * (1 + x)^3))^(n+1).

Mathematica

Table[SeriesCoefficient[((1+x)^2/(1-x*(1+x)^3))^(n+1), {x, 0, n}]/(n+1), {n, 0, 30}] (* Vincenzo Librandi, Oct 17 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x*(1+x)^3)/(1+x)^2)/x)
(Magma) [1/(n+1)*&+[Binomial(n+k, k)*Binomial(2*n+3*k+2, n-k): k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Oct 17 2025

Crossrefs

Cf. A062992, A121873, A274734.

Sequence in context: A135074 A292752 A220379 * A191800 A286764 A180609

Adjacent sequences: A387663 A387664 A387665 * A387667 A387668 A387669

Keyword

nonn

Author

Seiichi Manyama, Oct 04 2025

Status

approved