A387308 a(n) = Sum_{k=0..n} 2^k * binomial(n+3,k+3) * binomial(2*k+6,k+6).

1, 20, 270, 3100, 32711, 328440, 3195276, 30433800, 285604605, 2651696300, 24425110138, 223636254660, 2038173128355, 18508562948400, 167596683043032, 1514149108652880, 13654464563944377, 122951462526317700, 1105768912324277670, 9934852156019798700, 89186900539764803391

Offset: 0

Author

Seiichi Manyama, Aug 25 2025

Keywords

• nonn
• new

Links

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

Crossrefs

Cf. A331793, A387307.

Programs

Magma

[&+[2^k * Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025

Mathematica

Table[Sum[2^k*Binomial[n+3,k+3]*Binomial[2*k+6,k+6],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *)

PARI

a(n) = sum(k=0, n, 2^k*binomial(n+3, k+3)*binomial(2*k+6, k+6));

Formula

n*(n+6)*a(n) = (n+3) * (5*(2*n+5)*a(n-1) - 9*(n+2)*a(n-2)) for n > 1.

a(n) = Sum_{k=0..floor(n/2)} 4^k * 5^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).

a(n) = [x^n] (1+5*x+4*x^2)^(n+3).

E.g.f.: exp(5*x) * BesselI(3, 4*x) / 8, with offset 3.