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A067653 Denominators of the coefficients in exp(x/(1-x)) power series. 7
1, 1, 2, 6, 24, 40, 720, 5040, 4480, 362880, 3628800, 13305600, 479001600, 6227020800, 29059430400, 1307674368000, 1609445376000, 13173608448000, 6402373705728000, 121645100408832000, 810967336058880000, 4644631106519040000, 86461594444431360000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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 sub-domain 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

REFERENCES

K. Knopp "Theory and application of infinite series" Dover p. 547

O. Perron "Uber das infinitare Verhalten der koeffizienten einer gewissen Potenzreihe", Archiv d. math.u.Phys.(3), Vol. 22, pp. 329-340. 1914

H. Fischer, Eine Theorie komplexwertiger Abelscher Limitierungsmethoden (A theory of complex valued abelian limitation methods), Dissertation (1987), pp. 29-32.

K. Zeller, W. Beekmann, Theorie der Limitierungsverfahren, Springer-Verlag, Berlin (1970).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..450

Richard P. Brent, M. L. Glasser, Anthony J. Guttmann, A Conjectured Integer Sequence Arising From the Exponential Integral, arXiv:1812.00316 [math.NT], 2018.

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.

FORMULA

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

EXAMPLE

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 + ... .

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 *)

PROG

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

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

(Sage) [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

CROSSREFS

Cf. A067764.

Sequence in context: A257546 A274038 A143383 * A090755 A192196 A000496

Adjacent sequences:  A067650 A067651 A067652 * A067654 A067655 A067656

KEYWORD

nonn,frac

AUTHOR

Benoit Cloitre, Feb 03 2002

EXTENSIONS

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

STATUS

approved

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Last modified January 20 04:43 EST 2019. Contains 319323 sequences. (Running on oeis4.)