A389440 Expansion of (1/x) * Series_Reversion( x * (1 - x^3 * (1 + x)^2) / (1 + x) ).

1, 1, 1, 2, 8, 29, 89, 265, 853, 2949, 10349, 36037, 125693, 444229, 1591133, 5742771, 20813095, 75745059, 277025463, 1018045897, 3756150333, 13904480767, 51626986691, 192242833121, 717785838121, 2686625428555, 10078398664703, 37885690132494, 142691988840916

Offset

0,4

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+2*k+1,n-3*k).

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

Mathematica

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

Crossrefs

Cf. A198888, A215340, A389441.

Sequence in context: A373905 A365757 A378427 * A208973 A365761 A100477

Adjacent sequences: A389437 A389438 A389439 * A389441 A389442 A389443

Keyword

nonn

Author

Seiichi Manyama, Oct 04 2025

Status

approved