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Search: id:A005727
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| A005727 |
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n-th derivative of x^x at x=1. Also called Lehmer-Comtet numbers.
(Formerly M0868)
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+0
6
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| 1, 1, 2, 3, 8, 10, 54, -42, 944, -5112, 47160, -419760, 4297512, -47607144, 575023344, -7500202920, 105180931200, -1578296510400, 25238664189504, -428528786243904, 7700297625889920, -146004847062359040, 2913398154375730560
(list; graph; listen)
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OFFSET
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0,3
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, table at foot of page.
H. W. Gould, Rocky Mountain J. Math. 26(2) 1996.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
D. H. Lehmer, Numbers associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, p. 461.
G. H. Hardy, A Course of Pure Mathematics, 10th ed., Cambridge University Press, 1960, p. 428.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
Joerg Arndt, Fxtbook
G. H. Hardy, A Course of Pure Mathematics, Cambridge, The University Press, 1908.
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FORMULA
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For n>0, a(n)=sum(b(n, k), k=0..n), where b(n, k) is a Lehmer-Comtet number of the first kind (see A008296).
E.g.f.: (1+x)^(1+x). a(n) = Sum_{k=0..n} Stirling1(n, k)*A000248(k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 02 2003
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MATHEMATICA
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NestList[ Factor[ D[ #1, x ] ]&, x^x, n ] /. (x->1)
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PROGRAM
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(PARI) a(n)=if(n<0, 0, n!*polcoeff((1+x+x*O(x^n))^(1+x), n))
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CROSSREFS
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Cf. A005168. Row sums of A008296.
Sequence in context: A165153 A121989 A010786 this_sequence A118089 A084917 A134713
Adjacent sequences: A005724 A005725 A005726 this_sequence A005728 A005729 A005730
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KEYWORD
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sign,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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