A389629 Expansion of (1/x) * Series_Reversion( x * (1 - x)^2 * (1 - x^2 / (1 - x)^3) ).

1, 2, 8, 41, 237, 1471, 9573, 64450, 445140, 3136431, 22455917, 162904860, 1194831048, 8845525811, 66010749049, 496045713558, 3750351364616, 28507411174904, 217732027287758, 1670124738336036, 12860359694332056, 99374880851233747, 770340128361664973

Offset

0,2

Links

Vincenzo Librandi, 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(3*n+k+1,n-2*k).

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

Mathematica

Table[(1/(n+1))*Sum[Binomial[n+k, k]*Binomial[3*n+k+1, n-2*k], {k, 0, Floor[n/2]}], {n, 0, 25}] (* Vincenzo Librandi, Nov 09 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2*(1-x^2/(1-x)^3))/x)
(Magma) [(1/(n+1))*&+[Binomial(n+k, k)*Binomial(3*n+k+1, n-2*k): k in [0..Floor(n/2)]] : n in [0..30] ]; // Vincenzo Librandi, Nov 09 2025

Crossrefs

Cf. A151374, A236339, A365751.

Cf. A378467, A389345.

Sequence in context: A389437 A389548 A217362 * A381817 A294084 A381826

Adjacent sequences: A389626 A389627 A389628 * A389630 A389631 A389632

Keyword

nonn

Author

Seiichi Manyama, Oct 09 2025

Status

approved