A361738 - OEIS

%I #16 Mar 23 2023 15:31:43

%S 1,0,6,6,90,180,1770,5040,39690,140280,964656,3922380,24755346,

%T 110486376,660153780,3137330196,18103340970,89794566576,506892467796,

%U 2589310074780,14419819659960,75181803891480,415298937771900,2196704341517400,12078576672927570

%N Diagonal of rational function 1/(1 - (x^2 + y^2 + z^2 + x^3*y*z)).

%H Winston de Greef, <a href="/A361738/b361738.txt">Table of n, a(n) for n = 0..1328</a>

%F a(n) = Sum_{k=0..floor(n/2)} (3*k)!/k!^3 * binomial(k,n-2*k).

%F From _Vaclav Kotesovec_, Mar 23 2023: (Start)

%F Recurrence: (n-1)*n^2*a(n) = -(n-1)^2*n*a(n-1) + 3*(n-1)*(3*n - 4)*(3*n - 2)*a(n-2) + 18*(n-2)*(3*n^2 - 6*n + 1)*a(n-3) + 27*(n-3)*(n-2)*n*a(n-4).

%F a(n) ~ sqrt(3) * (6*cos(Pi/9))^n  / (2*Pi*n). (End)

%t Table[Sum[(3*k)!/k!^3 * Binomial[k,n-2*k], {k,0,n/2}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 23 2023 *)

%o (PARI) a(n) = sum(k=0, n\2, (3*k)!/k!^3*binomial(k, n-2*k));

%Y Cf. A361737, A361739.

%Y Cf. A361729.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 22 2023