A387278 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+1,k+1) * binomial(2*k+2,k+2).
1, 10, 78, 560, 3885, 26550, 180285, 1221400, 8272251, 56062550, 380361212, 2583867720, 17575724491, 119705522370, 816297170310, 5572945684800, 38088275031435, 260576833989150, 1784382167211378, 12229792774162800, 83888652677196591, 575858959975595010
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+1,k+1) * Binomial(2*k+2,k+2): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025
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Mathematica
Table[Sum[3^(n-k)*Binomial[n+1,k+1]*Binomial[2*k+2,k+2],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *) -
PARI
a(n) = sum(k=0, n, 3^(n-k)*binomial(n+1, k+1)*binomial(2*k+2, k+2));
Formula
n*(n+2)*a(n) = (n+1) * (5*(2*n+1)*a(n-1) - 21*n*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = [x^n] (1+5*x+x^2)^(n+1).
E.g.f.: exp(5*x) * BesselI(1, 2*x), with offset 1.