A067653 Denominators of the coefficients in exp(x/(1-x)) power series.

1, 1, 2, 6, 24, 40, 720, 5040, 4480, 362880, 3628800, 13305600, 479001600, 6227020800, 29059430400, 1307674368000, 1609445376000, 13173608448000, 6402373705728000, 121645100408832000, 810967336058880000, 4644631106519040000, 86461594444431360000

Offset: 0

Author

Benoit Cloitre, Feb 03 2002

Keywords

• nonn
• frac

Comments

Define c(n)=A067764(n)/A067653(n). For a given sequence s(n) consider P[s(n)](z):=e^(-z/(1-z))*Sum_{k>=0} s(k)c(k)z^k. Regarding complex valued abelian limitation the following holds true: if s(n) is convergent (to the limit s) then lim P[s(n)](z)=s as z tends to +1 in a certain subdomain D of the unit circle. There are two constraints: (1) D contains the line [0,1[. (2) There is a d>0 such that the intersection of {w|Re(w)>1-d} and D is a nonempty subset of a generalized Stolz set defined by {w||Im(w)|<=t*(1-Re(w))^(3/2)}, t<1. If z tends to +1 from outside such a domain that limit doesn't exist in general. - Hieronymus Fischer, Oct 20 2010

There is a significant overlap between the terms of this sequence and the terms of factorials (A000142): 44 of the first 100 terms of this sequence are also factorials. - Harvey P. Dale, Oct 27 2011

If there is no cancellation in the defining sum, a(n) = n!. In particular, a(p) = p! for prime p. In any case a(n) | n!. - Charles R Greathouse IV, Oct 27 2011

Examples

			exp(x/(1-x)) = 1 + x + (3/2)*x^2 + (13/6)*x^3 + (73/24)*x^4 + (167/40)*x^5 + (4051/720)*x^6 + (37633/5040)*x^7 + (43817/4480)*x^8 + (4596553/362880)*x^9 + ... .
		

References

• H. Fischer, Eine Theorie komplexwertiger Abelscher Limitierungsmethoden (A theory of complex valued abelian limitation methods), Dissertation (1987), pp. 29-32.
• K. Knopp, Theory and application of infinite series, Dover, p. 547.
• O. Perron, Über das infinitäre Verhalten der Koeffizienten einer gewissen Potenzreihe, Archiv d. Math. u. Phys. (3), Vol. 22, pp. 329-340, 1914.
• K. Zeller, W. Beekmann, Theorie der Limitierungsverfahren, Springer-Verlag, Berlin (1970).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450
• D. Borwein, On methods of summability based on power series, Proc. Royal Soc. Edinburgh, Sect. A Volume 64 / Issue 04 / January 1957, pp 342-349.
• Richard P. Brent, M. L. Glasser, and Anthony J. Guttmann, A Conjectured Integer Sequence Arising From the Exponential Integral, arXiv:1812.00316 [math.NT], 2018.

Crossrefs

Cf. A067764.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1-x)))); [Denominators(b[n]): n in [1..m]]; // G. C. Greubel, Dec 04 2018

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add((n-k)*b(k), k=0..n-1)/n)
    end:
a:= n-> denom(b(n)):
seq(a(n), n=0..25);  # Alois P. Heinz, May 12 2016

Mathematica

Denominator[Rest[CoefficientList[Series[Exp[x/(1-x)],{x,0,20}],x]]] (* Harvey P. Dale, Oct 26 2011 *)
r[n_] := If[n == 0, 1, Hypergeometric1F1[1 - n, 2, -1]]; Table[Denominator@  r[n], {n, 0, 22}] (* Peter Luschny, Feb 02 2019 *)

PARI

apply(x->denominator(x),Vec(exp(x/(1-x)))) \\ Charles R Greathouse IV, Oct 27 2011

SageMath

[denominator(sum(binomial(n-1, j-1)/factorial(j) for j in (1..n))) for n in range(30)] # G. C. Greubel, Dec 04 2018

Formula

a(n) is the denominator of Sum_{i=1..n} binomial(n-1, i-1)/i!.

Extensions

a(0)=1 prepended by Alois P. Heinz, May 12 2016