A372240 -id:A372240 - cp's OEIS frontend

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

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

Offset: 0

Vaclav Kotesovec, Feb 21 2015

nonn

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

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

Cf. A000178, A255358, A255359, A255360.
Cf. A002109, A051675, A255321, A255323, A255344.
Cf. A000142, A255268, A255269, A255504.
Cf. A272094, A372240, A372241.

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 *)

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

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)

A372241 a(n) = Product_{j=1..n} j^(ceiling(sqrt(j))).

Original entry on oeis.org

1, 1, 4, 36, 576, 72000, 15552000, 5334336000, 2731180032000, 1991030243328000, 19910302433280000000, 291506737925652480000000, 6044683717626329825280000000, 172642211659125606139822080000000, 6632223203096969285467405025280000000, 335756299656784070076787379404800000000000

Offset: 0

Vaclav Kotesovec, Apr 23 2024

nonn

Vaclav Kotesovec, Graph - the asymptotic ratio (1000 terms)

Cf. A255322, A372240.

Cf. A088021.

Table[Product[j^(Ceiling[Sqrt[j]]), {j, 1, n}], {n, 0, 15}]

a(n^2) = (n^2)!^(n+1) / A255322(n).
log(a(n)) ~ (2*n^(3/2)/3 + n/2 - sqrt(n)/6 + 1/4)*log(n) - 4*n^(3/2)/9 - n/2 + sqrt(n).
a(n^2) / A372240(n^2) = (n^2)! / n!^2 = A088021(n).

cp's OEIS Frontend

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

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