A361730 Diagonal of rational function 1/(1 - (1 + x*y*z) * (x^3 + y^3 + z^3)).
1, 0, 0, 6, 18, 18, 96, 540, 1350, 3480, 16470, 61020, 175860, 627480, 2498580, 8520876, 28563570, 106917300, 393495396, 1369171188, 4914119826, 18191218716, 65741140080, 235643531508, 862450963704, 3163777886412, 11484836808588, 41875694151720
Offset: 0
Keywords
Links
- Winston de Greef, Table of n, a(n) for n = 0..1723
Programs
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PARI
a(n) = sum(k=0, n\3, (3*k)!/k!^3*binomial(3*k, n-3*k));
Formula
a(n) = Sum_{k=0..floor(n/3)} (3*k)!/k!^3 * binomial(3*k,n-3*k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
Recurrence: (n-1)*n^2*a(n) = -(n-1)^2*n*a(n-1) + 27*(n-2)*(n-1)^2*a(n-3) + 108*(n-2)*(n^2 - 3*n + 1)*a(n-4) + 54*(3*n^3 - 18*n^2 + 28*n - 5)*a(n-5) + 108*(n^3 - 7*n^2 + 12*n - 1)*a(n-6) + 27*(n-5)*(n-3)*n*a(n-7).
a(n) ~ sqrt(3) * ((3 + sqrt(21))/2)^n / (2*Pi*n). (End)