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%I #15 Mar 23 2023 06:48:20
%S 1,0,0,6,6,0,90,180,90,1680,5040,5040,36330,138600,207900,895356,
%T 3818430,7567560,24720696,106702596,258053796,742135680,3050807760,
%U 8483450976,23450218506,89691647760,273414861720,760735601340,2713845780360,8733512193120,24957399366900
%N Diagonal of rational function 1/(1 - (x^3 + y^3 + z^3 + x^4*y*z)).
%H Winston de Greef, <a href="/A361739/b361739.txt">Table of n, a(n) for n = 0..1935</a>
%F a(n) = Sum_{k=0..floor(n/3)} (3*k)!/k!^3 * binomial(k,n-3*k).
%F From _Vaclav Kotesovec_, Mar 23 2023: (Start)
%F Recurrence: (n-1)*n^2*a(n) = -(n-1)^2*n*a(n-1) + 27*(n-2)*(n-1)^2*a(n-3) + 18*(n-2)*(3*n^2 - 9*n + 2)*a(n-4) + 3*n*(3*n - 11)*(3*n - 7)*a(n-5).
%F a(n) ~ sqrt(3) * d^n / (2*Pi*n), where d = 3.278393896770041178744966998018587... is the positive real root of the equation d^4 - 27*d - 27 = 0. (End)
%o (PARI) a(n) = sum(k=0, n\3, (3*k)!/k!^3*binomial(k, n-3*k));
%Y Cf. A361737, A361738.
%Y Cf. A361730.
%K nonn
%O 0,4
%A _Seiichi Manyama_, Mar 22 2023
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