A057528 5th level factorials: product of first n 4th level factorials.

1, 1, 2, 96, 31850496, 2524286414780230533120, 1189172215782988266980141580906985588465965465600000

Offset

0,3

Comments

In general for k-th level factorials a(n) =Product of first n (k-1)-th level factorials =Product[i^C(n-i+k-1,n-i)] over 1<=i<=n.

Links

Table of n, a(n) for n=0..6.

Formula

a(n) =a(n-1)*A057527(n) =Product[i^A000292(n-i+4)] over 1<=i<=n.

a(n) ~ exp(25/144 - 109*n/144 - 35*n^2/24 - 379*n^3/432 - 125*n^4/576 - 137*n^5/7200 + (35 + 30*n + 6*n^2)*Zeta(3)/(96*Pi^2) - Zeta(5)/(32*Pi^4) + (5+2*n)*Zeta'(-3)/12) * n^((5+2*n)*(19/288 + 25*n/144 + 5*n^2/36 + n^3/24 + n^4/240)) * (2*Pi)^((n+1)*(n+2)*(n+3)*(n+4)/48) / A^((5+2*n)*(5 + 5*n + n^2)/12), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-3) = A259068 = 0.00537857635777430114441697421... and A = A074962 = 1.282427129100622636875... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 24 2015

Mathematica

Table[Product[i^Binomial[n-i+4, 4], {i, 1, n}], {n, 0, 10}] (* Vaclav Kotesovec, Jul 24 2015 *)
Nest[FoldList[Times, #]&, Range[0, 10]!, 4] (* Harvey P. Dale, Dec 15 2021 *)

Crossrefs

Cf. A000142, A000178, A055462, A057527, A260404 for first, second, third, fourth and sixth level factorials.

Sequence in context: A091810 A344662 A165642 * A346565 A224986 A164335

Adjacent sequences: A057525 A057526 A057527 * A057529 A057530 A057531

Keyword

easy,nonn

Author

Henry Bottomley, Sep 02 2000

Status

approved