OFFSET
0,6
COMMENTS
Also constant term in the expansion of (1 + x^2 + y^2 + z^2 + 1/(x*y*z))^n.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = n! * Sum_{k=0..floor(n/5)} 1/(k!^3 * (2*k)! * (n-5*k)!) = Sum_{k=0..floor(n/5)} binomial(n,5*k) * A001460(k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
Recurrence: 2*n^3*(2*n - 5)*a(n) = 2*(10*n^4 - 40*n^3 + 50*n^2 - 30*n + 7)*a(n-1) - 10*(n-1)*(4*n^3 - 18*n^2 + 26*n - 13)*a(n-2) + 40*(n-2)^3*(n-1)*a(n-3) - 10*(n-3)*(n-2)*(n-1)*(2*n - 5)*a(n-4) + 3129*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ sqrt(c) * (1 + 5/2^(2/5))^n / (Pi^(3/2) * n^(3/2)), where c = 3.154712586460560795509193778252140601572145506226776094640234924884123818... is the real root of the equation -30634915689 + 95407210000*c - 127160000000*c^2 + 79846400000*c^3 - 25600000000*c^4 + 3276800000*c^5 = 0. (End)
MATHEMATICA
Table[n! * Sum[1/(k!^3 * (2*k)! * (n-5*k)!), {k, 0, n/5}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 22 2023 *)
PROG
(PARI) a(n) = n!*sum(k=0, n\5, 1/(k!^3*(2*k)!*(n-5*k)!));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 20 2023
STATUS
approved