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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| A variant of A000142, the factorial numbers. - N. J. A. Sloane (njas(AT)research.att.com), 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).
Sum_{n>=0} 1/a(n) = 1/e [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 03 2009]
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REFERENCES
| 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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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..160
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FORMULA
| 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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MATHEMATICA
| nn=20; CoefficientList[Series[1/(1+x), {x, 0, nn}], x]Range[0, nn]! (* or *) Join[{1}, RecurrenceTable[{a[1]==-1, a[n]==-n*a[n-1]}, a[n], {n, 20}]] (* From Harvey P. Dale, May 10 2011 *)
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PROG
| (PARI) {a(n) = if( n<0, 0, (-1)^n * n! )}
(MAGMA) [(-1)^n * Factorial(n): n in [0..25]]; // Vincenzo Librandi, May 12 2011
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CROSSREFS
| Cf. (-1)^n * A000142(n) = a(n).
Sequence in context: A124355 * A159333 A165233 A074166 A130641 A129655
Adjacent sequences: A133939 A133940 A133941 * A133943 A133944 A133945
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Sep 30 2007
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