A386919 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(2*n-k,n-k).

1, 6, 58, 624, 7050, 81926, 969640, 11624976, 140708682, 1715727090, 21043480458, 259331888712, 3208566672792, 39830312782344, 495853462219600, 6188170518911264, 77393543796042570, 969771226630919754, 12172039459124750062, 153006230384961477600, 1925930502301667496250

Offset: 0

Author

Seiichi Manyama, Aug 08 2025

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..350

Crossrefs

Cf. A066381, A386918, A386920.

Cf. A385639.

Programs

Magma

[&+[Binomial(4*n,k) * Binomial(2*n-k,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 10 2025

Mathematica

Table[Sum[Binomial[4*n,k]*Binomial[2*n-k,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 10 2025 *)

PARI

a(n) = sum(k=0, n, binomial(4*n, k)*binomial(2*n-k, n-k));

Formula

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

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

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

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