A391202 Expansion of (g/(1 - x*g^2))^3, where g = 1+x*g^4 is the g.f. of A002293.

1, 6, 36, 233, 1611, 11712, 88424, 687186, 5462931, 44224834, 363372606, 3022697811, 25407093194, 215463805722, 1841310131316, 15841150747579, 137089378343277, 1192579985966346, 10423026268316338, 91477851018404289, 805892537664730056, 7124020331497266292

Offset

0,2

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 A390709.

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

Mathematica

Table[Sum[(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 03 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (2*k+3)*binomial(k+2, 2)*binomial(4*n-2*k+3, n-k)/(4*n-2*k+3));
(Magma) [&+[(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 03 2025

Crossrefs

Cf. A390811, A391201.

Cf. A002293, A390709, A391199.

Sequence in context: A144819 A284647 A099252 * A057395 A259819 A213282

Adjacent sequences: A391199 A391200 A391201 * A391203 A391204 A391205

Keyword

nonn

Author

Seiichi Manyama, Dec 02 2025

Status

approved