A387403 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+3,k) * binomial(n+3,n-k), where i is the imaginary unit.
1, 8, 50, 280, 1484, 7616, 38304, 190080, 934560, 4564736, 22189024, 107476096, 519180480, 2502850560, 12046666752, 57912029184, 278136798720, 1334832967680, 6402435630080, 30695114813440, 147110418036736, 704860523102208, 3376580007936000, 16172904859238400
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
-
Magma
[&+[2^(n-k) * Binomial(n+3,n-2*k) * Binomial(2*k+3,k): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025
-
Mathematica
Table[Sum[2^(n-k)*Binomial[n+3,n-2*k]*Binomial[2*k+3,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+3, n-2*k)*binomial(2*k+3, k));
Formula
n*(n+6)*a(n) = (n+3) * (2*(2*n+5)*a(n-1) + 4*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+2*x+2*x^2)^(n+3).
E.g.f.: exp(2*x) * BesselI(3, 2*sqrt(2)*x) / (2*sqrt(2)), with offset 3.