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Search: id:A133942
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| 1, -1, 2, -6, 24, -120, 720, -5040, 40320, -362880, 3628800, -39916800, 479001600, -6227020800, 87178291200, -1307674368000, 20922789888000, -355687428096000, 6402373705728000, -121645100408832000, 2432902008176640000
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OFFSET
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0,3
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COMMENT
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A variant of A000142, the factorial numbers. - njas, Oct 03 2007
The terms of this sequences form the factorial series which Euler called the divergent series par excellence.
Euler summed this series to 0.596347... (A073003 = Gompertz's constant).
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REFERENCES
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V. S. Varadarajan, Euler and his Work on Infinite Series, Bull. Amer. Math. Soc., 44 (No. 4, 2007), 515-539. (See p. 527 and 530.)
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FORMULA
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Sum((-1)^i * i^n * binomial(n, i), i=0..n) = (-1)^n * n! - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
Stirling transform of a(n)=[1,-1,2,-6,24,...] is A000007(n)=[1,0,0,0,0,...].
a(n) = -n * a(n-1) unless n=0.
E.g.f.: 1/(1 + x).
G.f.: integral(t=1/x,infinity, (e^-t)/t) e^(1/x)/x = 1/(1 + x/(1 + x/(1 + 2x/(1 + 2x/(1 + 3x/(1 + 3x/(1 + ...))))))).
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PROGRAM
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(PARI) {a(n) = if( n<0, 0, (-1)^n * n! )}
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CROSSREFS
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Cf. (-1)^n * A000142(n) = a(n).
Adjacent sequences: A133939 A133940 A133941 this_sequence A133943 A133944 A133945
Sequence in context: A000142 A104150 A124355 this_sequence A074166 A130641 A129655
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Sep 30 2007
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