A388727 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(2*n+4*k,k).

1, 7, 83, 1120, 15955, 234257, 3506600, 53201624, 815208275, 12586819621, 195514618393, 3051863691125, 47830851950392, 752193577324066, 11863490726169508, 187578983467996610, 2972386405447826963, 47191255108524388845, 750512939573601611057, 11954068171154218220846

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Formula

a(n) = [x^n] ((1+x)^2 * (x+(1+x)^4))^n.

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / ((1+x)^2 * (x+(1+x)^4)) ). See A388731.

Mathematica

Table[Sum[ Binomial[n, k]*Binomial[2*n+4*k, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 22 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(2*n+4*k, k));
(Magma)
[&+[Binomial(n, k)*Binomial(2*n+4*k, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 22 2025

Crossrefs

Cf. A005809, A114496, A387105, A388725.

Cf. A388535, A388728, A388729.

Cf. A388731.

Sequence in context: A348303 A240170 A386920 * A231259 A396680 A383231

Adjacent sequences: A388724 A388725 A388726 * A388728 A388729 A388730

Keyword

nonn

Author

Seiichi Manyama, Sep 20 2025

Status

approved