A366305 -id:A366305 - cp's OEIS frontend

A366306 a(n) = Product_{k=1..n} (k^n - (k-1)^n).

1, 3, 133, 170625, 10733002621, 50465283999665535, 25145494699347449245677097, 1787473773567267792523164108726890625, 23480751910878672340765325385856840967957995534681, 71672834655019406921956925590632596034005848922160549420728589375

Offset: 1

Vaclav Kotesovec, Oct 06 2023

nonn

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

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

a(n) = (n!)^n * Product_{k=1..n} (1 - (1 - 1/k)^n).
a(n) ~ n!^n * d^n, where d = exp(Integral_{x=0..1} log(1 - exp(-1/x)) dx) = 0.84207793096051704199642805288991601369639823969574423397520945175552718...
a(n) ~ (2*Pi)^(n/2) * d^n * n^(n*(2*n+1)/2) / exp(n^2 - 1/12).

A366329 a(n) = Product_{k=1..n} Sum_{j=1..k} j^n.

Original entry on oeis.org

1, 5, 324, 589764, 52393770000, 347773153451938500, 244632735619259069507040000, 24547871392966749661547369532868031040, 455140097017244017295446005144727669016636127744000, 1960564895414510364772369567330640938816177001699555385515625000000

Offset: 1

Vaclav Kotesovec, Oct 07 2023

nonn

Cf. A031971, A036740, A366305, A366342.

Table[Product[Sum[j^n, {j, 1, k}], {k, 1, n}], {n, 1, 12}]
Table[Product[HarmonicNumber[k, -n], {k, 1, n}], {n, 1, 12}] // FunctionExpand

a(n) = A036740(n) * Product_{k=1..n} Sum_{j=1..k} (j/k)^n.
a(n) ~ n!^n * c * d^n, where d = exp(-Integral_{x=0..1} log(1 - exp(-1/x)) dx) = 1.187538543919977798892363400109897833660222697152558038684860736484... and c = exp(1 - 1/(exp(1) - 1)) / (exp(1) - 1) = 0.88399704290317414073109479991305699114875723090346..., updated Apr 19 2024
a(n) ~ c * d^n * (2*Pi)^(n/2) * n^(n*(2*n+1)/2) / exp(n^2 - 1/12).

cp's OEIS Frontend

A366306 a(n) = Product_{k=1..n} (k^n - (k-1)^n).

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