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A036740 a(n) = (n!)^n. 41
1, 1, 4, 216, 331776, 24883200000, 139314069504000000, 82606411253903523840000000, 6984964247141514123629140377600000000, 109110688415571316480344899355894085582848000000000, 395940866122425193243875570782668457763038822400000000000000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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 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

J. Borwein, D. Bailey and R. 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, arXiv:math/9902004 [math.CO], 1999.

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

a(n) = prod(k=1, n, (k-1)!*k^k); a(n) = A000178(n-1)*A002109(n). - Benoit Cloitre, Sep 17 2005

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

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 *)

PROG

(PARI) a(n)=n!^n;

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

CROSSREFS

A002109(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.

Cf. A086687, A225764, A261490.

Main diagonal of A225816.

Sequence in context: A055627 A260619 A167888 * A038786 A268362 A072694

Adjacent sequences:  A036737 A036738 A036739 * A036741 A036742 A036743

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 21 00:30 EST 2018. Contains 299388 sequences. (Running on oeis4.)