A394693 - OEIS

%I #38 May 08 2026 19:49:16

%S 0,1,85,69553,303434560,4933612230976,236578220012998656,

%T 28396321874429111046144,7558550239576242674347278336,

%U 4065590800747144716726929064984576,4105365186944970446848750402928640000000,7331569001911783916809480047177435709440000000

%N a(n) = (n!)^6 * Sum_{1<=i<=j<=k<=n} 1/(i*j*k)^2.

%H Seiichi Manyama, <a href="/A394693/b394693.txt">Table of n, a(n) for n = 0..104</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>

%F 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.

%F a(n) = 2 * (n!)^8 * Sum_{k=1..n} (-1)^(k-1)/(k^6 * (n-k)! * (n+k)!).

%F 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.

%F a(n) ~ 31 * Pi^9 * n^(6*n+3) / (1890 * exp(6*n)). - _Vaclav Kotesovec_, May 08 2026

%t 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 *)

%o (PARI) a(n) = 2*n!^8*sum(k=1, n, (-1)^(k-1)/(k^6*(n-k)!*(n+k)!));

%o (Python)

%o from math import factorial

%o from sympy import harmonic

%o 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

%Y Cf. A203229, A291456, A393984.

%K nonn

%O 0,3

%A _Seiichi Manyama_, May 08 2026