A389403 Expansion of (1/x) * Series_Reversion( x * (1 - x - x^2 / (1 - x)^3) ).

1, 1, 3, 13, 62, 311, 1625, 8769, 48509, 273606, 1567513, 9096623, 53361471, 315901789, 1884941061, 11324476869, 68446077906, 415898672925, 2539118226618, 15567603291314, 95812573306767, 591737630618789, 3666121030395939, 22779159645465623, 141911884921174169

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

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

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

Mathematica

Table[SeriesCoefficient[1/(1-x-x^2/(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-x^2/(1-x)^3))/x)
(Magma) [1/(n+1)*&+[Binomial(n+k, k)*Binomial(2*n+2*k, n-2*k): k in [0..Floor(n/2)]]: n in [0..30]]; // Vincenzo Librandi, Oct 17 2025

Crossrefs

Cf. A367415, A389402, A389404.

Cf. A001003, A378467.

Cf. A389345.

Sequence in context: A141786 A122122 A093424 * A391064 A186242 A367060

Adjacent sequences: A389400 A389401 A389402 * A389404 A389405 A389406

Keyword

nonn

Author

Seiichi Manyama, Oct 02 2025

Status

approved