A255322 a(n) = Product_{k=0..n} (k^2)!.

1, 1, 24, 8709120, 182219087869378560000, 2826438545846116156142906806150103040000000000, 1051416277636507481568264360276689674557030810000137484550133942059008000000000000000000

Offset: 0

Author

Vaclav Kotesovec, Feb 21 2015

Keywords

• nonn

Comments

Partial products of A088020. - Michel Marcus, Jul 06 2019

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..12

Crossrefs

Cf. A000178, A255358, A255359, A255360.

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

Cf. A000142, A255268, A255269, A255504.

Cf. A272094, A372240, A372241.

Programs

Mathematica

Table[Product[(k^2)!, {k, 0, n}], {n, 0, 10}]
FoldList[Times,(Range[0,6]^2)!] (* Harvey P. Dale, Jan 30 2022 *)
Table[(n^2)!^(n+1) / Product[j^(Ceiling[Sqrt[j]]), {j, 1, n^2}], {n, 0, 6}] (* Vaclav Kotesovec, Apr 23 2024 *)
Table[(n^2)!^n * (n!)^2 / Product[j^(Floor[Sqrt[j]]), {j, 1, n^2}], {n, 0, 6}] (* Vaclav Kotesovec, Apr 23 2024 *)

PARI

{a(n) = prod(k=1, n, (k^2)!)} \\ Seiichi Manyama, Jul 06 2019

Formula

a(n) ~ c * n^((2*n + 1)*(2*n^2 + 2*n + 3)/6) * (2*Pi)^(n/2) / exp(5*n^3/9 + n^2/2 + n), where c = A255504 = 3.048330306522348566911920417337613015885313475... .

From Vaclav Kotesovec, Apr 23 2024: (Start)

a(n) = Product_{j=1..n^2} j^(n - ceiling(sqrt(j)) + 1).

a(n) = (n^2)!^n * (n!)^2 / Product_{j=1..n^2} j^(floor(sqrt(j))). (End)