%I A002109 M3706 N1514
%S A002109 1,1,4,108,27648,86400000,4031078400000,3319766398771200000,55696437941726556979200000,
%T A002109 21577941222941856209168026828800000,215779412229418562091680268288000000000000000,
%U A002109 61564384586635053951550731889313964883968000000000000000
%N A002109 Hyperfactorials: Product_{k = 1..n} k^k.
%C A002109 a(n)=(-1)^n/det(M_n) where M_n is the n X n matrix m(i,j)=(-1)^i/i^j 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), May 28 2002
%C A002109 a(n) = determinant of the n X n matrix M(n) where m(i,j)=B(n,i,j) and 
               B(n,i,x) denote the Bernstein polynomial : B(n,i,x)=binomial(n,i)*(1-x)^(n-i)*x^i. 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 02 2003
%D A002109 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002109 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002109 Azarian, Mohammad K., On the hyperfactorial function, hypertriangular 
               function and the discriminants of certain polynomials. Int. J. Pure 
               Appl. Math. 36 (2007), 251-257.
%D A002109 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.
%D A002109 A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index 
               of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford 
               and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
%D A002109 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 1990, p. 477.
%H A002109 N. J. A. Sloane, <a href="b002109.txt">Table of n, a(n) for n = 1..36</
               a>
%H A002109 S. R. Finch, <a href="http://algo.inria.fr/bsolve/constant/glshkn/glshkn.html">
               Glaisher-Kinkelin Constant</a> (gives asymptotic expressions for 
               A002109, A000178) [At present this link does not work]
%H A002109 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Hyperfactorial.html">Link to a section of The World of Mathematics.</
               a>
%H A002109 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               K-Function.html">Link to a section of The World of Mathematics.</
               a>
%H A002109 <a href="Sindx_Fa.html#factorial">Index entries for sequences related 
               to factorial numbers</a>
%F A002109 Determinant of n X n matrix m(i, j)=binomial(i*j, i) - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Aug 27 2003
%p A002109 f := proc(n) local k; mul(k^k,k=1..n); end;
%p A002109 a[0]:=1:for n from 1 to 20 do a[n]:=product(n!/k!, k=0..n) od: seq(a[n], 
               n=0..11); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 11 
               2007
%p A002109 seq(mul(mul(k,j=1..k), k=1..n), n=0..11); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Sep 21 2007
%t A002109 lst={};s=1;Do[AppendTo[lst, s*=n^n], {n, 4!}];lst [From Vladimir Orlovsky 
               (4vladimir(AT)gmail.com), Sep 27 2008]
%t A002109 Table[Hyperfactorial[n], {n, 0, 11}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jul 10 2009]
%Y A002109 Cf. A000178, A000142.
%Y A002109 A002109(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.
%Y A002109 Cf. A001358, A002981, A002982, A100015, A005234, A014545, A018239, A006794, 
               A057704, A057705.
%Y A002109 Cf. A074962 [Glaisher-Kinkelin constant, also gives an asymptotic approximation 
               for the hyperfactorials].
%Y A002109 Sequence in context: A090205 A061464 A107048 this_sequence A076265 A114876 
               A037980
%Y A002109 Adjacent sequences: A002106 A002107 A002108 this_sequence A002110 A002111 
               A002112
%K A002109 nonn,easy,nice
%O A002109 0,3
%A A002109 N. J. A. Sloane (njas(AT)research.att.com).