A387402 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+2,k) * binomial(n+2,n-k), where i is the imaginary unit.
1, 6, 32, 160, 780, 3752, 17920, 85248, 404640, 1918400, 9090048, 43064320, 204032192, 966887040, 4583424000, 21735350272, 103114538496, 489392157696, 2323701678080, 11037970513920, 52454251902976, 249373626208256, 1186024281341952, 5642924625100800, 26858183388774400, 127880625111662592
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
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Magma
[&+[2^(n-k) * Binomial(n+2,n-2*k) * Binomial(2*k+2,k): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025
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Mathematica
Table[Sum[2^(n-k)*Binomial[n+2,n-2*k]*Binomial[2*k+2,k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, Sep 04 2025 *) -
PARI
a(n) = sum(k=0, n\2, 2^(n-k)*binomial(n+2, n-2*k)*binomial(2*k+2, k));
Formula
n*(n+4)*a(n) = (n+2) * (2*(2*n+3)*a(n-1) + 4*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+2*x+2*x^2)^(n+2).
E.g.f.: exp(2*x) * BesselI(2, 2*sqrt(2)*x) / 2, with offset 2.