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A002109
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Hyperfactorials: Product_{k = 1..n} k^k.
(Formerly M3706 N1514)
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22
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1, 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, 55696437941726556979200000, 21577941222941856209168026828800000, 215779412229418562091680268288000000000000000, 61564384586635053951550731889313964883968000000000000000
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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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).
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
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
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.
Partial products of A000312. - Reinhard Zumkeller, Jul 07 2012
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REFERENCES
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Azarian, Mohammad K., On the hyperfactorial function, hypertriangular function and the discriminants of certain polynomials. Int. J. Pure Appl. Math. 36 (2007), 251-257.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.
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.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 477.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
L. Toth, Weighted Gcd-Sum Functions, Journal of Integer Sequences, 14 (2011), #11.7.7.
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 1..36
S. R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [At present this link does not work]
Eric Weisstein's World of Mathematics, Hyperfactorial
Eric Weisstein's World of Mathematics, K-Function
Index to divisibility sequences
Index entries for sequences related to factorial numbers
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FORMULA
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Determinant of n X n matrix m(i, j)=binomial(i*j, i) - Benoit Cloitre, Aug 27 2003
log a(n) = 0.5 n^2 log n + n^2/4 + O(n log n). [Charles R Greathouse IV, Jan 12 2012]
a(n) = exp(zeta'(-1, n + 1) - zeta'(-1)) where zeta(s, z) is the Hurwitz zeta function. - Peter Luschny, Jun 23 2012
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MAPLE
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f := proc(n) local k; mul(k^k, k=1..n); end;
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
seq(mul(mul(k, j=1..k), k=1..n), n=0..11); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 21 2007
A002109 := n -> exp(Zeta(1, -1, n+1)-Zeta(1, -1)); seq(simplify(A002109(n)), n=0..11); # Peter Luschny, Jun 23 2012
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MATHEMATICA
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lst={}; s=1; Do[AppendTo[lst, s*=n^n], {n, 4!}]; lst [From Vladimir Joseph Stephan Orlovsky, Sep 27 2008]
Table[Hyperfactorial[n], {n, 0, 11}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]
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PROG
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(PARI) a(n)=prod(k=2, n, k^k) \\ Charles R Greathouse IV, Jan 12 2012
(Haskell)
a002109 n = a002109_list !! n
a002109_list = scanl1 (*) a000312_list -- Reinhard Zumkeller, Jul 07 2012
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CROSSREFS
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Cf. A000178, A000142.
A002109(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.
Cf. A001358, A002981, A002982, A100015, A005234, A014545, A018239, A006794, A057704, A057705, A054374.
Cf. A074962 [Glaisher-Kinkelin constant, also gives an asymptotic approximation for the hyperfactorials].
Sequence in context: A107048 A185702 A212803 * A076265 A114876 A037980
Adjacent sequences: A002106 A002107 A002108 * A002110 A002111 A002112
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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