A387274 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+4,k+4) * binomial(2*k+8,k+8).
1, 20, 246, 2408, 20636, 162288, 1203000, 8546208, 58823919, 395245708, 2606333730, 16933021560, 108703640136, 691068080928, 4358220121296, 27301946599872, 170074452183570, 1054434358722024, 6510869338671852, 40063301434583504, 245781459952640040
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..800
Programs
-
Magma
[&+[2^(n-k) * Binomial(n+4,k+4) * Binomial(2*k+8,k+8): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025
-
Mathematica
Table[Sum[2^(n-k)*Binomial[n+4,k+4]*Binomial[2*k+8,k+8],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *) -
PARI
a(n) = sum(k=0, n, 2^(n-k)*binomial(n+4, k+4)*binomial(2*k+8, k+8));
Formula
n*(n+8)*a(n) = (n+4) * (4*(2*n+7)*a(n-1) - 12*(n+3)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+4,n-2*k) * binomial(2*k+4,k).
a(n) = [x^n] (1+4*x+x^2)^(n+4).
E.g.f.: exp(4*x) * BesselI(4, 2*x), with offset 4.