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A053556
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Denominator of Sum_{k=0..n} (-1)^k/k!.
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24
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1, 1, 2, 3, 8, 30, 144, 280, 5760, 45360, 44800, 3991680, 43545600, 172972800, 6706022400, 93405312000, 42268262400, 22230464256000, 376610217984000, 250298560512000, 11640679464960000, 196503623737344000, 17841281393295360000
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OFFSET
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0,3
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COMMENTS
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Denominator of probability of a derangement of n things (A000166(n)/n!).
Also numerators 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
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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
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L. Euler, Introduction à l'analyse infinitésimale, Tome premier, Tome second, trad. du latin en français par J. B. Labey, Paris, 1796-1797.
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FORMULA
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EXAMPLE
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1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760, ...
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MATHEMATICA
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Table[Denominator[Sum[(-1)^k/k!, {k, 0, n}]], {n, 0, 20}] (* Robert G. Wilson v, Oct 13 2005 *)
Denominator[Accumulate[Table[(-1)^k/k!, {k, 0, 30}]]] (* Harvey P. Dale, Aug 22 2016 *)
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PROG
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(PARI) for(n=0, 50, print1(denominator(sum(k=0, n, (-1)^k/k!)), ", ")) \\ G. C. Greubel, Nov 05 2017
(Magma) [Denominator( (&+[(-1)^k/Factorial(k): k in [0..n]]) ): n in [0..20]]; // G. C. Greubel, May 16 2019
(Sage) [denominator(sum((-1)^k/factorial(k) for k in (0..n))) for n in (0..20)] # G. C. Greubel, May 16 2019
(Python)
from math import factorial
from fractions import Fraction
def A053556(n): return sum(Fraction(-1 if k&1 else 1, factorial(k)) for k in range(n+1)).denominator # Chai Wah Wu, Jul 31 2023
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CROSSREFS
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KEYWORD
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nonn,frac,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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