A387307 a(n) = Sum_{k=0..n} 2^k * binomial(n+2,k+2) * binomial(2*k+4,k+4).

1, 15, 166, 1650, 15615, 144025, 1309084, 11793780, 105663885, 943298675, 8401596258, 74716379270, 663813240363, 5894026429725, 52314876771960, 464261939106600, 4119843554861913, 36560929542771735, 324489293583792990, 2880380080564191450, 25572856871556696471

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, A387308.

Programs

Magma

[&+[2^k * Binomial(n+2,k+2) * Binomial(2*k+4,k+4): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025

Mathematica

Table[Sum[2^k*Binomial[n+2,k+2]*Binomial[2*k+4,k+4],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *)

PARI

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

Formula

n*(n+4)*a(n) = (n+2) * (5*(2*n+3)*a(n-1) - 9*(n+1)*a(n-2)) for n > 1.

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

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

E.g.f.: exp(5*x) * BesselI(2, 4*x) / 4, with offset 2.