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Search: id:A002109
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| A002109 |
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Hyperfactorials: Product_{k = 1..n} k^k.
(Formerly M3706 N1514)
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13
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| 1, 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, 55696437941726556979200000, 21577941222941856209168026828800000, 215779412229418562091680268288000000000000000, 61564384586635053951550731889313964883968000000000000000
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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
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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.
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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, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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 (benoit7848c(AT)orange.fr), Aug 27 2003
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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
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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.
Cf. A074962 [Glaisher-Kinkelin constant, also gives an asymptotic approximation for the hyperfactorials].
Sequence in context: A090205 A061464 A107048 this_sequence A076265 A122149 A114876
Adjacent sequences: A002106 A002107 A002108 this_sequence A002110 A002111 A002112
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas
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