A385668 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).

1, 13, 274, 6466, 160564, 4104733, 106927384, 2822352952, 75224906716, 2020064928916, 54569506803574, 1481263780787122, 40369492671395476, 1103922337550185894, 30274295947104877312, 832318570941153758356, 22932288741241396871068, 633044952458953424442364

Offset: 0

Author

Seiichi Manyama, Aug 04 2025

Keywords

• nonn

Links

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

Crossrefs

Cf. A385438.

Programs

Magma

[&+[2^(n-k) * Binomial(4*n+1,k) * Binomial(4*n-k,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 05 2025

Mathematica

Table[Sum[2^(n-k)*Binomial[4*n+1,k]*Binomial[4*n-k,n-k],{k,0,n}],{n,0,35}] (* Vincenzo Librandi, Aug 05 2025 *)

PARI

a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n+1, k)*binomial(4*n-k, n-k));

Formula

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

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

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

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

a(n) ~ 2^(8*n + 5/2) / (11 * sqrt(Pi*n) * 3^(2*n - 1/2)). - Vaclav Kotesovec, Aug 05 2025