A366306 - OEIS

%I #7 Oct 07 2023 08:48:36

%S 1,3,133,170625,10733002621,50465283999665535,

%T 25145494699347449245677097,1787473773567267792523164108726890625,

%U 23480751910878672340765325385856840967957995534681,71672834655019406921956925590632596034005848922160549420728589375

%N a(n) = Product_{k=1..n} (k^n - (k-1)^n).

%F a(n) = (n!)^n * Product_{k=1..n} (1 - (1 - 1/k)^n).

%F a(n) ~ n!^n * d^n, where d = exp(Integral_{x=0..1} log(1 - exp(-1/x)) dx) = 0.84207793096051704199642805288991601369639823969574423397520945175552718...

%F a(n) ~ (2*Pi)^(n/2) * d^n * n^(n*(2*n+1)/2) / exp(n^2 - 1/12).

%t Table[Product[k^n - (k-1)^n, {k, 1, n}], {n, 1, 10}]

%Y Cf. A036740, A323575, A323588, A323589, A366305.

%K nonn

%O 1,2

%A _Vaclav Kotesovec_, Oct 06 2023