0,3

(-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 (benoit7848c(AT)orange.fr), Sep 17 2005

Contribution from W. Edwin Clark (eclark(AT)math.usf.edu), Apr 09 2009: (Start)

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 by n matrix with entries from an n element set such that the

elements in each column (or row) are distinct. (End)

Christian Krattenthaler, Advanced Determinant Calculus.

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 (se16(AT)btinternet.com), Dec 06 2001

a(n)=prod(k=1, n, (k-1)!*k^k); a(n)=A000178(n-1)*A002109(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 17 2005

seq(mul(mul(j, j=1..n), k=1..n), n=0..9); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 02 2007

seq(mul(j^n, j=1..n), n=0..9); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 02 2007

restart:with (combinat):a:=n->mul(-stirling1(n, 1), j=2..n): seq(a(n), n=1..10); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 31 2008]

lst={}; Do[a=n!^n; AppendTo[lst, a], {n, 0, 13}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 01 2008]

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

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

Sequence in context: A042325 A091287 A055627 this_sequence A038786 A072694 A068210

Adjacent sequences: A036737 A036738 A036739 this_sequence A036741 A036742 A036743

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).