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A002109
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Hyperfactorials: Product_{k = 1..n} k^k.
(Formerly M3706 N1514)
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17
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1, 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, 55696437941726556979200000, 21577941222941856209168026828800000, 215779412229418562091680268288000000000000000, 61564384586635053951550731889313964883968000000000000000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
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
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REFERENCES
| 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
| 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
| Determinant of n X n matrix m(i, j)=binomial(i*j, i) - Benoit Cloitre (benoit7848c(AT)orange.fr), 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]
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MAPLE
| 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
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MATHEMATICA
| lst={}; s=1; Do[AppendTo[lst, s*=n^n], {n, 4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 27 2008]
Table[Hyperfactorial[n], {n, 0, 11}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]
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PROG
| (PARI) a(n)=prod(k=2, n, k^k) \\ Charles R Greathouse IV, Jan 12 2012
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CROSSREFS
| 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.
Cf. A074962 [Glaisher-Kinkelin constant, also gives an asymptotic approximation for the hyperfactorials].
Sequence in context: A061464 A107048 A185702 * A076265 A114876 A037980
Adjacent sequences: A002106 A002107 A002108 * A002110 A002111 A002112
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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