A387339 a(n) = Sum_{k=0..n} 3^k * binomial(n+2,k) * binomial(n+2,k+2).
1, 12, 108, 880, 6855, 52164, 391720, 2918304, 21634290, 159880600, 1179180552, 8685874080, 63930198787, 470327654580, 3459353475600, 25442360389696, 187126561024686, 1376455855989672, 10126540146288520, 74515694338112160, 548444877468906726
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..800
Programs
-
Magma
[&+[3^k * Binomial(n+2,k) * Binomial(n+2,k+2): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025
-
Mathematica
Table[Sum[3^k * Binomial[n+2,k]*Binomial[n+2, k+2],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *) -
PARI
a(n) = sum(k=0, n, 3^k*binomial(n+2, k)*binomial(n+2, k+2));
Formula
n*(n+4)*a(n) = (n+2) * (4*(2*n+3)*a(n-1) - 4*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+4*x+3*x^2)^(n+2).
E.g.f.: exp(4*x) * BesselI(2, 2*sqrt(3)*x) / 3, with offset 2.