A361658 - OEIS

%I #20 Mar 20 2023 07:32:34

%S 1,1,1,1,1,1,121,841,3361,10081,25201,55441,194041,1287001,7927921,

%T 38438401,152312161,516079201,1627691521,5745472321,25999820401,

%U 133086258481,651284938921,2860955078521,11312609403481,42039298455001,158864460354601,658342633033801

%N Constant term in the expansion of (1 + x^3 + y^3 + z^3 + 1/(x*y*z))^n.

%F a(n) = n! * Sum_{k=0..floor(n/6)} 1/(k!^3 * (3*k)! * (n-6*k)!) = Sum_{k=0..floor(n/6)} binomial(n,6*k) * A001421(k).

%F From _Vaclav Kotesovec_, Mar 20 2023: (Start)

%F Recurrence: (n-4)*(n-2)*n^3*a(n) = (6*n^5 - 45*n^4 + 112*n^3 - 123*n^2 + 68*n - 15)*a(n-1) - 3*(n-1)*(5*n^4 - 40*n^3 + 111*n^2 - 132*n + 59)*a(n-2) + 2*(n-2)*(n-1)*(10*n^3 - 75*n^2 + 181*n - 144)*a(n-3) - (n-3)*(n-2)*(n-1)*(15*n^2 - 90*n + 133)*a(n-4) + 3*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 7)*a(n-5) + 1727*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6).

%F a(n) ~ (1 + 2*sqrt(3))^(n + 3/2) / (4 * 3^(1/4) * Pi^(3/2) * n^(3/2)). (End)

%t Table[n!*Sum[1/(k!^3 * (3*k)! * (n-6*k)!), {k, 0, n/6}], {n, 0, 30}] (* _Vaclav Kotesovec_, Mar 20 2023 *)

%o (PARI) a(n) = n!*sum(k=0, n\6, 1/(k!^3*(3*k)!*(n-6*k)!));

%Y Cf. A002426, A361657.

%Y Cf. A001421, A361637.

%K nonn

%O 0,7

%A _Seiichi Manyama_, Mar 19 2023