A057527 - OEIS

%I #12 Jan 08 2024 14:08:19

%S 1,1,2,48,331776,79254226206720,471092427871945743012986880000,

%T 351177419973413722592573060611594181593855426560000000000

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

%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)*A055462(n) =Product[i^A000332(n-i)] over 1<=i<=n.

%F a(n) ~ exp(11/72 - 5*n/6 - 4*n^2/3 - 11*n^3/18 - 25*n^4/288 + Zeta(3)*(n+2) / (8*Pi^2) + Zeta'(-3)/6) * n^(251/720 + n + 11*n^2/12 + n^3/3 + n^4/24) * (2*Pi)^((n+1)*(n+2)*(n+3)/12) / A^(11/6 + 2*n + n^2/2), where Zeta(3) = A002117, Zeta'(-3) = A259068 = 0.0053785763577743011444169742104138428956644397... and A = A074962 = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Jul 24 2015

%e a(4) =((4!*3!*2!*1!)*(3!*2!*1!)*(2!*1!)*(1!)) * ((3!*2!*1!)*(2!*1!)*(1!)) * ((2!*1!)*(1!)) * ((1!)) =24*6^3*2^6*1^10 =331776

%t Table[Product[i^Binomial[n-i+3,3],{i,1,n}],{n,0,10}] (* _Vaclav Kotesovec_, Jul 24 2015 *)

%t Nest[FoldList[Times,#]&,Range[0,8]!,3] (* _Harvey P. Dale_, Jan 08 2024 *)

%Y Cf. A000142, A000178, A055462, A057528, A260404 for first, second, third, fifth and sixth level factorials.

%K easy,nonn

%O 0,3

%A _Henry Bottomley_, Sep 02 2000