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5th level factorials: product of first n 4th level factorials.
6

%I #9 Dec 15 2021 14:26:23

%S 1,1,2,96,31850496,2524286414780230533120,

%T 1189172215782988266980141580906985588465965465600000

%N 5th level factorials: product of first n 4th level factorials.

%C 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.

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

%F 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

%t Table[Product[i^Binomial[n-i+4,4],{i,1,n}],{n,0,10}] (* _Vaclav Kotesovec_, Jul 24 2015 *)

%t Nest[FoldList[Times,#]&,Range[0,10]!,4] (* _Harvey P. Dale_, Dec 15 2021 *)

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

%K easy,nonn

%O 0,3

%A _Henry Bottomley_, Sep 02 2000