A387311 a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k+3) * binomial(2*k+6,k+6).

1, 28, 535, 8750, 132041, 1900808, 26557986, 363716220, 4912064355, 65673861484, 871539802276, 11501122783696, 151118588963615, 1978948331160080, 25846338449608184, 336857447941007280, 4382848524348689883, 56947000383926523780, 739095412895790074215, 9583718189242229830798

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. A387309, A387310.

Programs

Magma

[&+[3^k * Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025

Mathematica

Table[Sum[3^k * Binomial[n+3,k+3]*Binomial[2*k+6, k+6],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)

PARI

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

Formula

n*(n+6)*a(n) = (n+3) * (7*(2*n+5)*a(n-1) - 13*(n+2)*a(n-2)) for n > 1.

a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).

a(n) = [x^n] (1+7*x+9*x^2)^(n+3).

E.g.f.: exp(7*x) * BesselI(3, 6*x) / 27, with offset 3.