

A036740


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


42



1, 1, 4, 216, 331776, 24883200000, 139314069504000000, 82606411253903523840000000, 6984964247141514123629140377600000000, 109110688415571316480344899355894085582848000000000, 395940866122425193243875570782668457763038822400000000000000000000
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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 sumofpowers 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(n1)*n^n*(n1)! = a(n1)*A000169(n)*A000142(n) = A036740(n1) * A000312(n)*A000142(n1).  Henry Bottomley, Dec 06 2001
a(n) = prod(k=1, n, (k1)!*k^k); a(n) = A000178(n1)*A002109(n).  Benoit Cloitre, Sep 17 2005
a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^21/12).  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(n1) = (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



