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A053557
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Numerator of Sum_{k=0..n} (-1)^k/k!.
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13
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1, 0, 1, 1, 3, 11, 53, 103, 2119, 16687, 16481, 1468457, 16019531, 63633137, 2467007773, 34361893981, 15549624751, 8178130767479, 138547156531409, 92079694567171, 4282366656425369, 72289643288657479, 6563440628747948887, 39299278806015611311
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Numerator of probability of a derangement of n things (A000166(n)/n! or !n/n!).
Also denominators of successive convergents to e using continued fraction 2+1/(1+1/(2+2/(3+3/(4+4/(5+5/(6+6/(7+7/8+...))))))).
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REFERENCES
| L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 562.
E. Maor, e: The Story of a Number, Princeton Univ. Press 1994, pp. 151 and 157.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
Leonhardo Eulero, Introductio in analysin infinitorum. Tomus primus, Lausanne, 1748.
L. Euler, Introduction a l'analyse infinitesimal Tome premier, Tome second, trad. du latin en francais par J. B. Labey, Paris, 1796-1797.
Eric Weisstein's World of Mathematics, Subfactorial [From Eric W. Weisstein (eric(AT)weisstein.com), Sep 12 2008]
Eric Weisstein's World of Mathematics, Continued Fraction Constant [From Eric W. Weisstein (eric(AT)weisstein.com), Oct 03 2008]
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FORMULA
| Let -e^(-x)/(x-1)= Sum(n, 0, infty)a_n/b_n *x^n . Then sequence a_n is A053557 - Aleksandar Petojevic (apetoje(AT)ptt.yu), Apr 14 2004
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EXAMPLE
| 1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760, ...
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MATHEMATICA
| Numerator[ CoefficientList[ Series[E^-x/(1-x), {x, 0, 23}], x]] (* From Jean-François Alcover, Nov 18 2011 *)
Table[Numerator[Sum[(-1)^k/k!, {k, 0, n}]], {n, 0, 30}] (* From Harvey P. Dale, Dec 02 2011 *)
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CROSSREFS
| Cf. A000166/A000142, A053556 (denominators), A053518-A053520. See also A103816.
a(n) = (N(n, n) of A103361), A053557/A053556 = A000166/n! = (N(n, n) of A103361)/(D(n, n) of A103360), Cf. A053518-A053520.
Sequence in context: A054362 A129833 A107958 * A039302 A074512 A005502
Adjacent sequences: A053554 A053555 A053556 * A053558 A053559 A053560
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KEYWORD
| nonn,frac,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jan 17 2000
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 31 2000
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