A387275 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+2,k+2) * binomial(2*k+4,k+4).
1, 15, 154, 1350, 10890, 83650, 623056, 4547520, 32735085, 233369675, 1652203542, 11638252730, 81674873553, 571575363975, 3991529920440, 27829484027400, 193791573179883, 1348196149698885, 9372495529924710, 65120144658997050, 452263192928596896
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+2,k+2) * Binomial(2*k+4,k+4): 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+2,k+2]*Binomial[2*k+4,k+4],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *) -
PARI
a(n) = sum(k=0, n, 3^(n-k)*binomial(n+2, k+2)*binomial(2*k+4, k+4));
Formula
n*(n+4)*a(n) = (n+2) * (5*(2*n+3)*a(n-1) - 21*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+5*x+x^2)^(n+2).
E.g.f.: exp(5*x) * BesselI(2, 2*x), with offset 2.
a(n) ~ 7^(n + 5/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 31 2025