A387276 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+3,k+3) * binomial(2*k+6,k+6).
1, 20, 255, 2650, 24521, 210840, 1725234, 13631700, 104993955, 793367300, 5907885412, 43495473840, 317355930255, 2298888740400, 16555878011448, 118661449810320, 847132614218907, 6027874235210700, 42773816956415055, 302816249208061050, 2139537520524710691
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..800
Programs
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Magma
[&+[3^(n-k) * Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025
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Mathematica
Table[Sum[3^(n-k)*Binomial[n+3,k+3]*Binomial[2*k+6,k+6],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *) -
PARI
a(n) = sum(k=0, n, 3^(n-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) - 21*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+5*x+x^2)^(n+3).
E.g.f.: exp(5*x) * BesselI(3, 2*x), with offset 3.