A387277 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+4,k+4) * binomial(2*k+8,k+8).
1, 25, 381, 4585, 47978, 458010, 4100370, 35027850, 288845370, 2317794050, 18203687502, 140533725150, 1069904389008, 8052575725680, 60033791987424, 444015014417280, 3261950250436845, 23827019766988725, 173193081555808545, 1253583401573658925, 9040278899072328006
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+4,k+4) * Binomial(2*k+8,k+8): 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+4,k+4]*Binomial[2*k+8,k+8],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *) -
PARI
a(n) = sum(k=0, n, 3^(n-k)*binomial(n+4, k+4)*binomial(2*k+8, k+8));
Formula
n*(n+8)*a(n) = (n+4) * (5*(2*n+7)*a(n-1) - 21*(n+3)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+4,n-2*k) * binomial(2*k+4,k).
a(n) = [x^n] (1+5*x+x^2)^(n+4).
E.g.f.: exp(5*x) * BesselI(4, 2*x), with offset 4.