A389404 Expansion of (1/x) * Series_Reversion( x * (1 - x - x^3 / (1 - x)^3) ).

1, 1, 2, 6, 23, 97, 422, 1866, 8383, 38286, 177564, 834476, 3965112, 19014625, 91898946, 447166731, 2188807224, 10770408900, 53247311424, 264359334110, 1317478927431, 6588555916430, 33052207924908, 166286792032456, 838802666233355, 4241470346653380

Offset

0,3

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(2*n+k,n-3*k).

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

Mathematica

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

Crossrefs

Cf. A367415, A389402, A389403.

Cf. A054515, A367413.

Sequence in context: A111283 A150297 A374165 * A150298 A391522 A395133

Adjacent sequences: A389401 A389402 A389403 * A389405 A389406 A389407

Keyword

nonn

Author

Seiichi Manyama, Oct 02 2025

Status

approved