A387338 a(n) = Sum_{k=0..n} 2^k * binomial(n+3,k) * binomial(n+3,k+3).
1, 12, 100, 720, 4809, 30744, 191184, 1167120, 7033785, 41999364, 249075684, 1469561184, 8636441905, 50600529840, 295755641152, 1725379046496, 10050215851665, 58470232877820, 339832224226180, 1973538115293360, 11453616812552761, 66436765880135112
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..800
Programs
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Magma
[&+[2^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[2^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, 2^k*binomial(n+3, k)*binomial(n+3, k+3));
Formula
n*(n+6)*a(n) = (n+3) * (3*(2*n+5)*a(n-1) - (n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 2^k * 3^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+3*x+2*x^2)^(n+3).
E.g.f.: exp(3*x) * BesselI(3, 2*sqrt(2)*x) / (2*sqrt(2)), with offset 3.