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A103816
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Numerator of Sum_{k=1..n} (-1)^(k+1)/k!.
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14
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0, 1, 1, 2, 5, 19, 91, 177, 3641, 28673, 28319, 2523223, 27526069, 109339663, 4239014627, 59043418019, 26718637649, 14052333488521, 238063061452591, 158218865944829, 7358312808534631, 124213980448686521, 11277840764547411113, 67527236643922308689
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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The Aitken delta-squared process leaves the sequence S(n) = Sum_{k=1..n} (-1)^(k+1)/k! essentially unchanged: S(n+3) = (S(n)*S(n+2) - (S(n+1))^2)/(S(n) + S(n+2) - 2*S(n+1)).
Numerators of coefficients in expansion of (1 - exp(-x)) / (1 - x). - Ilya Gutkovskiy, May 24 2022
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EXAMPLE
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0, 1, 1/2, 2/3, 5/8, 19/30, 91/144, 177/280, 3641/5760, 28673/45360, 28319/44800, ...
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MAPLE
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b:= proc(n) b(n):=`if`(n<2, 1-n, (n-1)*(b(n-1)+b(n-2))) end:
a:= n-> numer((n!-b(n))/n!):
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MATHEMATICA
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Join[{0}, Accumulate[Times@@@Partition[Riffle[1/Range[30]!, {1, -1}, {2, -1, 2}], 2]]//Numerator] (* Harvey P. Dale, Apr 18 2023 *)
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PROG
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(Python)
from math import factorial
from fractions import Fraction
def A103816(n): return sum(Fraction(1 if k&1 else -1, factorial(k)) for k in range(1, n+1)).numerator # 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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