A387340 a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k) * binomial(n+3,k+3).
1, 16, 175, 1640, 14189, 117152, 939036, 7379040, 57188010, 438810592, 3342302821, 25316084248, 190937278805, 1435287936320, 10760879892008, 80509920297792, 601343784616830, 4485466826475360, 33420579148668670, 248788060638391120, 1850652536242372786
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..800
Programs
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Magma
[&+[3^k * Binomial(n+3,k) * Binomial(n+3,k+3): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025
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Mathematica
Table[Sum[3^k * Binomial[n+3,k]*Binomial[n+3, k+3],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *) -
PARI
a(n) = sum(k=0, n, 3^k*binomial(n+3, k)*binomial(n+3, k+3));
Formula
n*(n+6)*a(n) = (n+3) * (4*(2*n+5)*a(n-1) - 4*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+4*x+3*x^2)^(n+3).
E.g.f.: exp(4*x) * BesselI(3, 2*sqrt(3)*x) / (3*sqrt(3)), with offset 3.