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 A002109 Hyperfactorials: Product_{k = 1..n} k^k. (Formerly M3706 N1514) 22

%I M3706 N1514

%S 1,1,4,108,27648,86400000,4031078400000,3319766398771200000,

%T 55696437941726556979200000,21577941222941856209168026828800000,

%U 215779412229418562091680268288000000000000000,61564384586635053951550731889313964883968000000000000000

%N Hyperfactorials: Product_{k = 1..n} k^k.

%C Comment from Alan Sokal, Mar 02 2012: A054374 gives the discriminants of the Hermite polynomials in the conventional (physicists') normalization, and A002109 gives the discriminants of the Hermite polynomials in the (in my opinion more natural) probabilists' normalization. See http://en.wikipedia.org/wiki/Hermite_polynomials and Szego, Orthogonal Polynomials, eq. (6.71.7).

%C 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_, May 28 2002

%C 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_, Feb 02 2003

%C The least significant non-zero digit of a(n): 1, 1, 4, 8, 8, 4, 4, 2, 2, 8, 8, 8, 8, 4, 4, 6, 6, 2, 8, 2, 2, 2, 8, 6, 6, 4, 4, 2, 2, 8, ... - _Robert G. Wilson v_, May 11 2012.

%C Partial products of A000312. - _Reinhard Zumkeller_, Jul 07 2012

%D 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 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.

%D 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 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 477.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D L. Toth, Weighted Gcd-Sum Functions, Journal of Integer Sequences, 14 (2011), #11.7.7.

%H N. J. A. Sloane, <a href="/A002109/b002109.txt">Table of n, a(n) for n = 1..36</a>

%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/glshkn/glshkn.html">Glaisher-Kinkelin Constant</a> (gives asymptotic expressions for A002109, A000178) [At present this link does not work]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hyperfactorial.html">Hyperfactorial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/K-Function.html">K-Function</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%F Determinant of n X n matrix m(i, j)=binomial(i*j, i) - _Benoit Cloitre_, Aug 27 2003

%F log a(n) = 0.5 n^2 log n + n^2/4 + O(n log n). [_Charles R Greathouse IV_, Jan 12 2012]

%F a(n) = exp(zeta'(-1, n + 1) - zeta'(-1)) where zeta(s, z) is the Hurwitz zeta function. - _Peter Luschny_, Jun 23 2012

%p f := proc(n) local k; mul(k^k,k=1..n); end;

%p 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 seq(mul(mul(k,j=1..k), k=1..n), n=0..11); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 21 2007

%p A002109 := n -> exp(Zeta(1,-1,n+1)-Zeta(1,-1)); seq(simplify(A002109(n)),n=0..11); # _Peter Luschny_, Jun 23 2012

%t lst={};s=1;Do[AppendTo[lst, s*=n^n], {n, 4!}];lst [From _Vladimir Joseph Stephan Orlovsky_, Sep 27 2008]

%t Table[Hyperfactorial[n], {n, 0, 11}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]

%o (PARI) a(n)=prod(k=2,n,k^k) \\ _Charles R Greathouse IV_, Jan 12 2012

%o a002109 n = a002109_list !! n

%o a002109_list = scanl1 (*) a000312_list -- _Reinhard Zumkeller_, Jul 07 2012

%Y Cf. A000178, A000142.

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

%Y Cf. A001358, A002981, A002982, A100015, A005234, A014545, A018239, A006794, A057704, A057705, A054374.

%Y Cf. A074962 [Glaisher-Kinkelin constant, also gives an asymptotic approximation for the hyperfactorials].

%K nonn,easy,nice,changed

%O 0,3

%A _N. J. A. Sloane_.

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