A389348 Expansion of (1/x) * Series_Reversion( x * (1 - x^3 / (1 - x)^2) ).

1, 0, 0, 1, 2, 3, 8, 23, 56, 139, 374, 1010, 2698, 7311, 20100, 55484, 153752, 428788, 1202070, 3382424, 9552176, 27073089, 76972788, 219454986, 627319364, 1797580445, 5162435342, 14856438822, 42836064422, 123732249059, 357999397548, 1037436676745, 3010782544074, 8749786321429

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(n-k-1,n-3*k).

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

Mathematica

Table[SeriesCoefficient[1/(1-x^3/(1-x)^2)^(n+1), {x, 0, n}]/(n+1), {n, 0, 30}] (* Vincenzo Librandi, Oct 18 2025 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1-x^3/(1-x)^2))/x)
(Magma) [1/(n+1)*&+[Binomial(n+k, k)*Binomial(n-k-1, n-3*k): k in [0..Floor(n/3)]]: n in [0..35]]; // Vincenzo Librandi, Oct 18 2025

Crossrefs

Cf. A389249.

Sequence in context: A006796 A241904 A006076 * A263459 A261061 A086628

Adjacent sequences: A389345 A389346 A389347 * A389349 A389350 A389351

Keyword

nonn

Author

Seiichi Manyama, Oct 01 2025

Status

approved