A387310 a(n) = Sum_{k=0..n} 3^k * binomial(n+2,k+2) * binomial(2*k+4,k+4).
1, 21, 330, 4690, 63690, 844662, 11052496, 143462592, 1852852365, 23853938185, 306473670822, 3932435239278, 50417223635233, 646085510253645, 8277409340709240, 106037993391958936, 1358437551566242347, 17404555385537336583, 223025734596708637750, 2858460480570547144110
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..800
Programs
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Magma
[&+[3^k * Binomial(n+2,k+2) * Binomial(2*k+4,k+4): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025
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Mathematica
Table[Sum[3^k * Binomial[n+2,k+2]*Binomial[2*k+4, k+4],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *) -
PARI
a(n) = sum(k=0, n, 3^k*binomial(n+2, k+2)*binomial(2*k+4, k+4));
Formula
n*(n+4)*a(n) = (n+2) * (7*(2*n+3)*a(n-1) - 13*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+7*x+9*x^2)^(n+2).
E.g.f.: exp(7*x) * BesselI(2, 6*x) / 9, with offset 2.