A255344 -id:A255344 - cp's OEIS frontend

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

1, 1, 263130836933693530167218012160000000

Offset: 0

Vaclav Kotesovec, Feb 21 2015

nonn

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).

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

Cf. A000178, A255322, A255358, A255359.
Cf. A002109, A051675, A255321, A255323, A255344.
Cf. A000142, A255268, A255269, A255439.

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 *)

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

A330716 n-th Gosper hyperfactorial of n.

Original entry on oeis.org

1, 1, 16, 1952152956156672

Offset: 0

Greg Huber, Dec 27 2019

Gosper's m-th hyperfactorial of n is the product 1^(1^m)*2^(2^m)*3^(3^m)*...*n^(n^m).

The 0th hyperfactorial is the factorial function.

			n=3: a(3) = 1^(1^3)*2^(2^3)*3^(3^3) = 2^8 * 3^27.
a(4) has 198 decimal digits and a(5) has 2927 digits.

R. W. Gosper, "Fac Fun" (ca. 1979).

R. W. Gosper, Gosper's facfun document

Cf. A000142, A002109, A051675, A255321, A255323, A255344 (0th through 5th Gosper hyperfactorials, respectively).

nmax:=3; Table[Product[i^(i^n),{i,1,n}],{n,0,nmax}] (* Stefano Spezia, Dec 29 2019 *)

A255403 Product_{k=1..n} (k^k)!.

Original entry on oeis.org

1, 24, 261332866810040451858432000000

Offset: 1

Vaclav Kotesovec, Feb 22 2015

nonn

The next term (a(4)) has 537 digits.

Vaclav Kotesovec, Table of n, a(n) for n = 1..4

Cf. A000178, A255322, A255358, A255359, A255360.
Cf. A002109, A051675, A255321, A255323, A255344.
Cf. A000142, A255268, A255269.

Table[Product[(k^k)!, {k, 1, n}], {n, 1, 4}]

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

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A330716 n-th Gosper hyperfactorial of n.

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A255403 Product_{k=1..n} (k^k)!.

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