A067764 Numerators of the coefficients in exp(x/(1-x)) power series.

1, 1, 3, 13, 73, 167, 4051, 37633, 43817, 4596553, 58941091, 274691047, 12470162233, 202976401213, 1178339174801, 65573803186921, 99264170666917, 994319127823939, 588633468315403843, 13564373693588558173, 109232642628695218147, 752832094524169066031

Offset: 0

Author

Benoit Cloitre, Feb 03 2002

Keywords

• nonn
• frac

Comments

Define c(n) = a(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_{z->+1} P[s(n)](z)=s 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

The ratio sequence given by c(n) = a(n)/A067653(n) also occurs in certain row and column sums related to Pascal's triangle, as in the two formulas given below. - Richard R. Forberg, Dec 26 2013

Examples

			Example for first formula. 1/1! + 3/2! + 3/3! + 1/4! = 73/24.
Example for 2nd formula. A000332 = 0, 0, 0, 0, 1, 5, 15, 35, 70, 126, ...; a(4) = 0/0! + 1/1! + 5/2! + 15/3! + 35/4! + 70/5! + 126/6! + ... = 73*e/24.
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

• O. Perron, Über das infinitäre 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..445
• D. Borwein, On methods of summability based on power series, Proc. Royal Soc. Edinburgh, Sect. A, Vol. 64(04), Jan. 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.
• K. Knopp, Theory and Application of Infinite Series, Dover, 1954, p. 547.

Crossrefs

Cf. A067653.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1-x)))); [Numerator(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-> numer(b(n)):
seq(a(n), n=0..25);  # Alois P. Heinz, May 12 2016

Mathematica

Table[Numerator@ SeriesCoefficient[Exp[x/(1 - x)], {x, 0, n}], {n, 19}] (* Michael De Vlieger, Dec 14 2015 *)
r[n_] := If[n == 0, 1, Hypergeometric1F1[1 - n, 2, -1]]; Table[Numerator@ r[n], {n, 0, 21}] (* Peter Luschny, Feb 02 2019 *)

PARI

a(n) = numerator(sum(k=1, n, binomial(n-1, k-1)/k!)); \\ Altug Alkan, Dec 14 2015

SageMath

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

Formula

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

a(n) is also the numerator of (Sum_{m>=0} binomial(n+m-1,n)/m!)/e, with A067653(n) as the denominator. See as example A000332 = binomial(n,4) below. - Richard R. Forberg, Dec 26 2013

a(n) = numerator(hypergeom([1 - n], [2], -1)) for n > 0. - Peter Luschny, Feb 02 2019

Extensions

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