A386870 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+2,k) * binomial(4*n-k,n-k).

1, 14, 297, 7024, 174608, 4466622, 116403982, 3073417652, 81935130444, 2200645300312, 59455990356377, 1614089892481416, 43993649464273588, 1203123469832767556, 32997093202771098204, 907229481990010791100, 24997561841045998756604, 690088514785377393552360

Offset: 0

Author

Seiichi Manyama, Aug 06 2025

Keywords

• nonn

Links

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

Crossrefs

Cf. A385668, A386845.

Programs

Magma

[&+[2^(n-k)*Binomial(4*n+2,k) * Binomial(4*n-k,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 09 2025

Mathematica

Table[Sum[2^(n-k)*Binomial[4*n+2,k]*Binomial[4*n-k,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 09 2025 *)

PARI

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

Formula

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

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

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

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