OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..104
Eric Weisstein's World of Mathematics, Harmonic Number
FORMULA
a(n) = (n!)^6 * (H(n,2)^3 + 3*H(n,2)*H(n,4) + 2*H(n,6))/6 where H(n,r) = Sum_{k=1..n} 1/k^r.
a(n) = 2 * (n!)^8 * Sum_{k=1..n} (-1)^(k-1)/(k^6 * (n-k)! * (n+k)!).
a(n) = n^2 * (3*n^4-6*n^3+7*n^2-4*n+1) * a(n-1) - n^2 * (n-1)^8 * (3*n^2-6*n+5) * a(n-2) + n^2 * (n-1)^8 * (n-2)^8 * a(n-3) + ((n-1)!)^6 for n > 2.
a(n) ~ 31 * Pi^9 * n^(6*n+3) / (1890 * exp(6*n)). - Vaclav Kotesovec, May 08 2026
MATHEMATICA
Table[n!^6 * (HarmonicNumber[n, 2]^3 + 3*HarmonicNumber[n, 2]*HarmonicNumber[n, 4] + 2*HarmonicNumber[n, 6])/6, {n, 0, 15}] (* Vaclav Kotesovec, May 08 2026 *)
PROG
(PARI) a(n) = 2*n!^8*sum(k=1, n, (-1)^(k-1)/(k^6*(n-k)!*(n+k)!));
(Python)
from math import factorial
from sympy import harmonic
def A394693(n): return int(factorial(n)**6*((m:=harmonic(n, 2))*(m**2+3*harmonic(n, 4))+2*harmonic(n, 6))/6) # Chai Wah Wu, May 08 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 08 2026
STATUS
approved