A388050 a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * binomial(4*n+1,k).

1, 16, 379, 9946, 273769, 7747048, 223222141, 6514324144, 191915445349, 5695383188728, 170004224419846, 5098613585824684, 153515140847257771, 4637565653158509136, 140494652062207134328, 4266717591819290005216, 129854662220912455666141, 3959496825621434094706624

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Formula

a(n) = [x^n] (1-2*x)^n/(1-3*x)^(4*n+2).

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

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

a(n) = [x^n] (1+x)^(4*n+1) * (3+x)^n.

Mathematica

Table[Sum[3^k*Binomial[n, k]*Binomial[4*n+1, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Sep 19 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 3^k*binomial(n, k)*binomial(4*n+1, k));
(Magma) [&+[3^k*Binomial(n, k)*Binomial(4*n+1, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 19 2025

Crossrefs

Cf. A388047, A388048, A388049.

Cf. A387932.

Sequence in context: A264073 A333581 A201617 * A235672 A235729 A189766

Adjacent sequences: A388047 A388048 A388049 * A388051 A388052 A388053

Keyword

nonn

Author

Seiichi Manyama, Sep 14 2025

Status

approved