A361729 - OEIS

%I #20 Mar 17 2024 08:42:43

%S 1,0,6,18,108,546,3030,16920,96480,557460,3255426,19186020,113905386,

%T 680583708,4088506428,24677473884,149564145060,909784736388,

%U 5552109174084,33981183515664,208523253915306,1282621025382840,7906367632595328,48832556909752044

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

%H Vaclav Kotesovec, <a href="/A361729/b361729.txt">Table of n, a(n) for n = 0..1230</a>

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

%F From _Vaclav Kotesovec_, Mar 22 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) + 12*(9*n^3 - 36*n^2 + 41*n - 9)*a(n-3) + 18*(3*n - 8)*(3*n^2 - 7*n + 1)*a(n-4) + 12*(9*n^3 - 54*n^2 + 80*n - 5)*a(n-5) + 3*n*(3*n - 13)*(3*n - 8)*a(n-6).

%F a(n) ~ c * d^n / n, where d = 6.45021022459140188868150633620495776554217848977385402261531271... is the real root of the equation -27 - 81*d - 81*d^2 - 27*d^3 + d^5 = 0 and c = sqrt(3)/(2*Pi) = 0.275664447710896024755663249156484720698693240183320326399... (End)

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

%Y Cf. A361728, A361730.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 22 2023