A387368 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(n+1,k) * binomial(n+1,n-k).
1, 10, 93, 860, 7985, 74550, 699685, 6597400, 62457921, 593346050, 5653702637, 54012503220, 517192500721, 4962377183470, 47698928343285, 459224987322800, 4427611044899585, 42744433267222650, 413145666547033213, 3997556929553596300, 38718094094951086641
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
Programs
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Magma
[&+[2^k * 3^(n-k) * Binomial(n+1,k) * Binomial(n+1,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+1,k]*Binomial[n+1, 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+1, k)*binomial(n+1, n-k));
Formula
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(n+1,k) * binomial(n+1,n-k).
n*(n+2)*a(n) = (n+1) * (5*(2*n+1)*a(n-1) - n*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 6^k * 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = [x^n] (1+5*x+6*x^2)^(n+1).
E.g.f.: exp(5*x) * BesselI(1, 2*sqrt(6)*x) / sqrt(6), with offset 1.