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 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.
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) ~ 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
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved