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A106827
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Numerators in expansion of (1 - x)^(-1/x) / e.
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2
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1, 1, 11, 315, 17129, 1510425, 196385475, 35327367075, 8399994587985, 2550903574364145, 963207568455370875, 442613044315692124875, 243195136160954426677305, 157442856285298191126143625, 118607799383105394973766029875, 102867257381973743111023517821875
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OFFSET
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0,3
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REFERENCES
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L. Comtet, Analyse Combinatoire, P. U. F., 1970, tome second, p. 140, #12.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, Problem 11.
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1 . 3 . 1.
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LINKS
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FORMULA
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Sum_{n>=0} a(n)/(2n)!*x^n = (1 - x)^(-1/x) / e.
a(n) = (-1)^n * Sum_{k=0..n} Stirling1(n+k, k) * !(n-k) * C(2*n, n+k), where !n = A000166(n) is the subfactorial, C(n,k) are binomial coefficients. - Vladimir Reshetnikov, Sep 23 2016
a(n) = (2*n)! * coefficients of Product_{j >= 2} exp(x^(j-1)/j). - G. C. Greubel, Sep 14 2021
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EXAMPLE
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G.f. = 1 + 1*x/2! + 11*x^2/4! + 315*x^3/6! + 17129*x^4/8! + 503475*x^5/10! + ...
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MATHEMATICA
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Table[(-1)^n Sum[StirlingS1[n+k, k] Subfactorial[n-k] Binomial[2n, n+k], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 23 2016 *)
With[{m=30}, CoefficientList[Series[(1-x)^(-1/x)/E, {x, 0, m}], x]*(2*Range[0, m])!] (* G. C. Greubel, Sep 14 2021 *)
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PROG
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(Magma) m:=31; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&*[Exp(x^(j-1)/j): j in [2..40]]) )); [Factorial(2*n-2)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 14 2021
(Sage)
def A_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( product(exp(x^(j-1)/j) for j in (2..41)) ).list()
A=A_list(40)
[factorial(2*n)*A[n] for n in (0..31)] # G. C. Greubel, Sep 14 2021
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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