A036740 a(n) = (n!)^n.

1, 1, 4, 216, 331776, 24883200000, 139314069504000000, 82606411253903523840000000, 6984964247141514123629140377600000000, 109110688415571316480344899355894085582848000000000, 395940866122425193243875570782668457763038822400000000000000000000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

(-1)^n*a(n) is the determinant of the n X n matrix m_{i,j} = T(n+i,j), 1 <= i,j <= n, where T(n,k) are the signed Stirling numbers of the first kind (A008275). Derived from methods given in Krattenthaler link. - Benoit Cloitre, Sep 17 2005

a(n) is also the number of binary operations on an n-element set which are right (or left) cancellative. These are also called right (left) cancellative magma or groupoids. The multiplication table of a right (left) cancellative magma is an n X n matrix with entries from an n element set such that the elements in each column (or row) are distinct. - W. Edwin Clark, Apr 09 2009

This sequence is mentioned in "Experimentation in Mathematics" as a sum-of-powers determinant. - John M. Campbell, May 07 2011

Determinant of the n X n matrix M_n = [m_n(i,j)] with m_n(i,j) = Stirling2(n+i,j) for 1<=i,j<=n. - Alois P. Heinz, Jul 26 2013

References

• Jonathan Borwein, David Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Ltd., 2004, p. 207.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..30
• Christian Krattenthaler, Advanced determinant calculus, in: D. Foata and G. N. Han (eds.), The Andrews Festschrift, Springer, Berlin, Heidelberg, 2001, pp. 349-426; arXiv preprint, arXiv:math/9902004 [math.CO], 1999.

Crossrefs

Cf. A000142, A000169, A000312.

Cf. A086687, A225764, A261114, A261490.

Main diagonal of A225816.

Programs

Maple

a:= n-> n!^n:
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 25 2013

Mathematica

Table[(n!)^n,{n,0,10}] (* Harvey P. Dale, Sep 29 2013 *)

Maxima

makelist(n!^n,n,0,10); /* Martin Ettl, Jan 13 2013 */

PARI

a(n)=n!^n;

Formula

a(n) = a(n-1)*n^n*(n-1)! = a(n-1)*A000169(n)*A000142(n) = A036740(n-1) * A000312(n)*A000142(n-1). - Henry Bottomley, Dec 06 2001

From Benoit Cloitre, Sep 17 2005: (Start)

a(n) = Product_{k=1..n} (k-1)!*k^k;

a(n) = A000178(n-1)*A002109(n) for n >= 1. (End)

a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2-1/12). - Vaclav Kotesovec, Nov 14 2014

a(n) = Product_{k=1..n} k^n. - José de Jesús Camacho Medina, Jul 12 2016

Sum_{n>=0} 1/a(n) = A261114. - Amiram Eldar, Nov 16 2020