cp's OEIS Frontend

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

%I A036740 #74 Sep 24 2024 15:56:59
%S A036740 1,1,4,216,331776,24883200000,139314069504000000,
%T A036740 82606411253903523840000000,6984964247141514123629140377600000000,
%U A036740 109110688415571316480344899355894085582848000000000,395940866122425193243875570782668457763038822400000000000000000000
%N A036740 a(n) = (n!)^n.
%C A036740 (-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
%C A036740 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
%C A036740 This sequence is mentioned in "Experimentation in Mathematics" as a sum-of-powers determinant. - _John M. Campbell_, May 07 2011
%C A036740 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
%D A036740 Jonathan Borwein, David Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Ltd., 2004, p. 207.
%H A036740 Alois P. Heinz, <a href="/A036740/b036740.txt">Table of n, a(n) for n = 0..30</a>
%H A036740 Christian Krattenthaler, <a href="https://doi.org/10.1007/978-3-642-56513-7_17">Advanced determinant calculus</a>, in: D. Foata and G. N. Han (eds.), The Andrews Festschrift, Springer, Berlin, Heidelberg, 2001, pp. 349-426; <a href="http://arxiv.org/abs/math/9902004">arXiv preprint</a>, arXiv:math/9902004 [math.CO], 1999.
%F A036740 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
%F A036740 From _Benoit Cloitre_, Sep 17 2005: (Start)
%F A036740 a(n) = Product_{k=1..n} (k-1)!*k^k;
%F A036740 a(n) = A000178(n-1)*A002109(n) for n >= 1. (End)
%F A036740 a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2-1/12). - _Vaclav Kotesovec_, Nov 14 2014
%F A036740 a(n) = Product_{k=1..n} k^n. - _José de Jesús Camacho Medina_, Jul 12 2016
%F A036740 Sum_{n>=0} 1/a(n) = A261114. - _Amiram Eldar_, Nov 16 2020
%p A036740 a:= n-> n!^n:
%p A036740 seq(a(n), n=0..12);  # _Alois P. Heinz_, Jul 25 2013
%t A036740 Table[(n!)^n,{n,0,10}] (* _Harvey P. Dale_, Sep 29 2013 *)
%o A036740 (PARI) a(n)=n!^n;
%o A036740 (Maxima) makelist(n!^n,n,0,10); /* _Martin Ettl_, Jan 13 2013 */
%Y A036740 Cf. A000142, A000169, A000312.
%Y A036740 Cf. A086687, A225764, A261114, A261490.
%Y A036740 Main diagonal of A225816.
%K A036740 nonn,easy
%O A036740 0,3
%A A036740 _N. J. A. Sloane_