A387273 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+3,k+3) * binomial(2*k+6,k+6).
1, 16, 165, 1400, 10661, 75936, 517524, 3420960, 22123530, 140782048, 885008839, 5511579528, 34073731965, 209428887360, 1281220578936, 7808422173120, 47440778110398, 287490594872160, 1738463164498410, 10493677382085744, 63245915436539682, 380697445274657984
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..800
Programs
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Magma
[&+[2^(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[2^(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, 2^(n-k)*binomial(n+3, k+3)*binomial(2*k+6, k+6));
Formula
n*(n+6)*a(n) = (n+3) * (4*(2*n+5)*a(n-1) - 12*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+4*x+x^2)^(n+3).
E.g.f.: exp(4*x) * BesselI(3, 2*x), with offset 3.