A391198 Expansion of g/(1 - x*g^2)^3, where g = 1+x*g^4 is the g.f. of A002293.

1, 4, 19, 107, 678, 4651, 33710, 254225, 1975179, 15703801, 127164198, 1045202460, 8697646655, 73134066873, 620427087397, 5303795766259, 45643712699236, 395114853630036, 3438140753061607, 30056496290696458, 263852525918990551, 2324963943767748904

Offset

0,2

Links

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

Formula

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

Mathematica

Table[Sum[(2*k+1)*Binomial[k+2, 2]*Binomial[4*n-2*k+1, n-k]/(4*n-2*k+1), {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Dec 03 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (2*k+1)*binomial(k+2, 2)*binomial(4*n-2*k+1, n-k)/(4*n-2*k+1));
(Magma) [&+[(2*k+1)*Binomial(k+2, 2) * Binomial(4*n-2*k+1, n-k)/(4*n-2*k+1): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Dec 03 2025

Crossrefs

Cf. A002293, A390237, A390709.

Sequence in context: A052751 A367284 A249934 * A174992 A182541 A377506

Adjacent sequences: A391195 A391196 A391197 * A391199 A391200 A391201

Keyword

nonn

Author

Seiichi Manyama, Dec 02 2025

Status

approved