login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A055462 Superduperfactorials: product of first n superfactorials. 17
1, 1, 2, 24, 6912, 238878720, 5944066965504000, 745453331864786829312000000, 3769447945987085350501386572267520000000000, 6916686207999802072984424331678589933649915805696000000000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Next term has 92 digits and is too large to display.

Starting with offset 1, a(n) is a 'Matryoshka doll' sequence with alpha=1, the multiplicative counterpart to the additive A000332. The sequence for m with alpha<=m<=L is then computed as Prod_{n=alpha..m}(Prod_{k=alpha..n}(Prod_{i=alpha..k}(i))). - Peter Luschny, Jul 14 2009

LINKS

Miles Seventh, Table of n, a(n) for n = 0..19

FORMULA

a(n) = a(n-1)*A000178(n) = Product[(i!)^(n-i+1)] over 1 <= i <= n = Product[i^((n-i+1)(n-i+2)/2)] over 1 <= i <= n.

log a(n) = (1/6) n^3 log n - (11/36) n^3 + O(n^2 log n). - Charles R Greathouse IV, Jan 13 2012

a(n) = exp((6 + 13 n + 9 n^2 + 2 n^3 - 8*(n + 2)*log(A)-2*(n + 2)^2*log(2*Pi) + 4*(2 n + 1)*logG(n + 2) - 4*(n + 1)^2*logGamma(n + 2) + 8*psi(-3, n + 2))/8) where A is the Glaisher-Kinkelin constant, logG(z) is the logarithm of the Barnes G function (A000178), and psi(-3, z) is a polygamma function of negative order (i.e., the second iterated integral of logGamma(z)). - Jan Mangaldan, Mar 21 2013

a(n) ~ exp(Zeta(3)/(8*Pi^2) - (2*n+3)*(11*n^2 + 24*n - 3)/72) * n^((2*n+3)*(2*n^2 + 6*n + 3)/24) * (2*Pi)^((n+1)*(n+2)/4) / A^(n+3/2), where A = A074962 = 1.28242712910062263687... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.2020569031595942853997... . - Vaclav Kotesovec, Feb 20 2015

EXAMPLE

a(4) = 1!2!3!4!*1!2!3!*1!2!*1! = 288*12*2*1 = 6912.

MAPLE

seq(mul(mul(mul(i, i=alpha..k), k=alpha..n), n=alpha..m), m=alpha..10); # Peter Luschny, Jul 14 2009

MATHEMATICA

s1=1; s2=1; lst={}; Do[f=n!; s1*=f; s2*=s1; AppendTo[lst, s2], {n, 0, 3*3!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jun 13 2009 *)

Table[Product[BarnesG[j], {j, k + 1}], {k, 10}] (* Jan Mangaldan, Mar 21 2013 *)

Table[Round[Exp[(n+2)*(n+3)*(2*n+5)/8] * Exp[PolyGamma[-3, n+3]] * BarnesG[n+3]^(n+3/2) / (Glaisher^(n+3) * (2*Pi)^((n+3)^2/4) * Gamma[n+3]^((n+2)^2/2))], {n, 0, 10}] (* Vaclav Kotesovec, Feb 20 2015 after Jan Mangaldan *)

PROG

(PARI) a(n)=my(t=1); prod(k=2, n, t*=k!) \\ Charles R Greathouse IV, Jul 28 2011

CROSSREFS

Cf. A000142, A000178, A002109, A036740, A255268, A255269.

Sequence in context: A242484 A088912 A203465 * A088600 A066120 A152687

Adjacent sequences:  A055459 A055460 A055461 * A055463 A055464 A055465

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, Jun 26 2000

EXTENSIONS

a(9) from N. J. A. Sloane, Dec 15 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified June 26 23:21 EDT 2017. Contains 288777 sequences.