A106827 - OEIS

%I #11 May 16 2023 10:52:50

%S 1,1,11,315,17129,1510425,196385475,35327367075,8399994587985,

%T 2550903574364145,963207568455370875,442613044315692124875,

%U 243195136160954426677305,157442856285298191126143625,118607799383105394973766029875,102867257381973743111023517821875

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

%D L. Comtet, Analyse Combinatoire, P. U. F., 1970, tome second, p. 140, #12.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, Problem 11.

%D S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1 . 3 . 1.

%H G. C. Greubel, <a href="/A106827/b106827.txt">Table of n, a(n) for n = 0..201</a>

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

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

%F 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

%F a(n) = (2*n)! * coefficients of Product_{j >= 2} exp(x^(j-1)/j). - _G. C. Greubel_, Sep 14 2021

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

%t 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 *)

%t With[{m=30}, CoefficientList[Series[(1-x)^(-1/x)/E, {x,0,m}], x]*(2*Range[0,m])!] (* _G. C. Greubel_, Sep 14 2021 *)

%o (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

%o (Sage)

%o def A_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( product(exp(x^(j-1)/j) for j in (2..41)) ).list()

%o A=A_list(40)

%o [factorial(2*n)*A[n] for n in (0..31)] # _G. C. Greubel_, Sep 14 2021

%Y Cf. A055505, A055535.

%K nonn,frac

%O 0,3

%A _Philippe Deléham_, May 21 2005

%E a(5) corrected by _G. C. Greubel_, Sep 14 2021