A255360 Product_{k=0..n} (k^5)!.

1, 1, 263130836933693530167218012160000000

Offset: 0

Author

Vaclav Kotesovec, Feb 21 2015

Keywords

• nonn

Comments

The next term a(3) has 512 digits.

In general (for m>1), product_{k=0..n} (k^m)! ~ c(m) * (2*Pi)^(n/2) * n^(m*(1/4 + n/2 + B(m+1)/(m+1) + (sum_{j=1..n} j^m) )) * exp(-m*n/2 - m*n^(m+1)/(m+1)^2 - (sum_{j=1..n} j^m) + m * (sum_{j=1..m-1} 1/(j+1) * B(j+1) * binomial(m, j) * n^(m-j) * (sum_{i=0..j-1} 1/(m-i)) )), where c(m) is a constant and B(n) is the Bernoulli number A027641(n)/A027642(n).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..3

Crossrefs

Cf. A000178, A255322, A255358, A255359.

Cf. A002109, A051675, A255321, A255323, A255344.

Cf. A000142, A255268, A255269, A255439.

Programs

Mathematica

Table[Product[(k^5)!, {k, 0, n}], {n, 0, 4}]
Table[Product[j^(n - Ceiling[j^(1/5)] + 1), {j, 1, n^5}], {n, 0, 4}] (* Vaclav Kotesovec, Apr 25 2024 *)

Formula

a(n) ~ c * n^(80/63 + 5*n/2 - 5*n^2/12 + 25*n^4/12 + 5*n^5/2 + (5*n^6)/6) * (2*Pi)^(n/2) / exp(5*n/2 + 35*n^2/144 + n^5/2 + 11*n^6/36), where c = A255439 = 11.354954749729782312106... .

a(n) = Product_{j=1..n^5} j^(n - ceiling(j^(1/5)) + 1). - Vaclav Kotesovec, Apr 25 2024