A106827 Numerators in expansion of (1 - x)^(-1/x) / e.

1, 1, 11, 315, 17129, 1510425, 196385475, 35327367075, 8399994587985, 2550903574364145, 963207568455370875, 442613044315692124875, 243195136160954426677305, 157442856285298191126143625, 118607799383105394973766029875, 102867257381973743111023517821875

Offset: 0

Author

Philippe Deléham, May 21 2005

Keywords

• nonn
• frac

Examples

			G.f. = 1 + 1*x/2! + 11*x^2/4! + 315*x^3/6! + 17129*x^4/8! + 503475*x^5/10! + ...
		

References

• 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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..201

Crossrefs

Cf. A055505, A055535.

Programs

Magma

m:=31; R:=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

Mathematica

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 *)

Sage

def A_list(prec):
    P. = 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

Formula

Sum_{n>=0} a(n)/(2n)!*x^n = (1 - x)^(-1/x) / e.

a(n) = A055505(n)*(2n)! / A055535(n).

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

Extensions

a(5) corrected by G. C. Greubel, Sep 14 2021