A387369 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(n+2,k) * binomial(n+2,n-k).
1, 15, 174, 1850, 18915, 189525, 1877596, 18476820, 181083285, 1770245675, 17278828842, 168496597230, 1642259489143, 16002398658225, 155919866646840, 1519307275471400, 14806582620440553, 144329229195062535, 1407215890063071910, 13724133021646678050, 133885448856624266571
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..800
Programs
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Magma
[&+[2^k * 3^(n-k) * Binomial(n+2,k) * Binomial(n+2,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025
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Mathematica
Table[Sum[2^k * 3^(n-k)*Binomial[n+2,k]*Binomial[n+2, n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *) -
PARI
a(n) = sum(k=0, n, 2^k*3^(n-k)*binomial(n+2, k)*binomial(n+2, n-k));
Formula
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(n+2,k) * binomial(n+2,n-k).
n*(n+4)*a(n) = (n+2) * (5*(2*n+3)*a(n-1) - (n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 6^k * 5^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+5*x+6*x^2)^(n+2).
E.g.f.: exp(5*x) * BesselI(2, 2*sqrt(6)*x) / 6, with offset 2.