A391609 Expansion of (g/(1 + x*g^2))^3, where g = 1+x*g^4 is the g.f. of A002293.

1, 0, 6, 33, 231, 1680, 12722, 99174, 790695, 6417782, 52854210, 440569659, 3709925170, 31512765744, 269690266146, 2323196695551, 20128333751349, 175286209221924, 1533444732404226, 13469998501882377, 118760989817062512, 1050602774154917980, 9322513339131716088

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

G.f.: B(x)^3, where B(x) is the g.f. of A390740.

a(n) = Sum_{k=0..n} (-1)^k * (2*k+3) * binomial(k+2,2) * binomial(4*n-2*k+3,n-k)/(4*n-2*k+3).

Mathematica

Table[Sum[(-1)^k*(2*k+3)*Binomial[k+2, 2]*Binomial[4*n-2*k+3, n-k]/(4*n-2*k+3), {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Dec 17 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*(2*k+3)*binomial(k+2, 2)*binomial(4*n-2*k+3, n-k)/(4*n-2*k+3));
(Magma) [&+[(-1)^k*(2*k+3)*Binomial(k+2, 2) * Binomial(4*n-2*k+3, n-k)/(4*n-2*k+3): k in [0..n]] : n in [0..30] ]; //  Vincenzo Librandi, Dec 17 2025

Crossrefs

Cf. A002293, A390740, A391202.

Sequence in context: A112903 A320420 A215707 * A137970 A024079 A228640

Adjacent sequences: A391606 A391607 A391608 * A391610 A391611 A391612

Keyword

nonn

Author

Seiichi Manyama, Dec 14 2025

Status

approved