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A255322
a(n) = Product_{k=0..n} (k^2)!.
17
1, 1, 24, 8709120, 182219087869378560000, 2826438545846116156142906806150103040000000000, 1051416277636507481568264360276689674557030810000137484550133942059008000000000000000000
OFFSET
0,3
COMMENTS
Partial products of A088020. - Michel Marcus, Jul 06 2019
LINKS
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)
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 *)
PROG
(PARI) {a(n) = prod(k=1, n, (k^2)!)} \\ Seiichi Manyama, Jul 06 2019
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Feb 21 2015
STATUS
approved