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%I #53 Dec 17 2023 17:37:12
%S 1,1,3,13,73,167,4051,37633,43817,4596553,58941091,274691047,
%T 12470162233,202976401213,1178339174801,65573803186921,99264170666917,
%U 994319127823939,588633468315403843,13564373693588558173,109232642628695218147,752832094524169066031
%N Numerators of the coefficients in exp(x/(1-x)) power series.
%C 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
%C 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
%D O. Perron, Über das infinitäre Verhalten der Koeffizienten einer gewissen Potenzreihe, Archiv d. Math. u. Phys. (3), Vol. 22, pp. 329-340, 1914.
%D H. Fischer, Eine Theorie komplexwertiger Abelscher Limitierungsmethoden (A theory of complex valued abelian limitation methods), Dissertation (1987), pp. 29-32.
%D K. Zeller, W. Beekmann, Theorie der Limitierungsverfahren, Springer-Verlag, Berlin (1970).
%H Alois P. Heinz, <a href="/A067764/b067764.txt">Table of n, a(n) for n = 0..445</a>
%H D. Borwein, <a href="http://dx.doi.org/10.1017/S0080454100007561">On methods of summability based on power series</a>, Proc. Royal Soc. Edinburgh, Sect. A, Vol. 64(04), Jan. 1957, pp 342-349.
%H Richard P. Brent, M. L. Glasser, and Anthony J. Guttmann, <a href="https://arxiv.org/abs/1812.00316">A Conjectured Integer Sequence Arising From the Exponential Integral</a>, arXiv:1812.00316 [math.NT], 2018.
%H K. Knopp, <a href="https://archive.org/details/theoryandapplica031692mbp/page/n561">Theory and Application of Infinite Series</a>, Dover, 1954, p. 547.
%F a(n) is the numerator of Sum_{i=1..n} binomial(n-1, i-1)/i!.
%F 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
%F a(n) = numerator(hypergeom([1 - n], [2], -1)) for n > 0. - _Peter Luschny_, Feb 02 2019
%e Example for first formula. 1/1! + 3/2! + 3/3! + 1/4! = 73/24.
%e 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.
%e 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 + ... .
%p b:= proc(n) option remember; `if`(n=0, 1,
%p add((n-k)*b(k), k=0..n-1)/n)
%p end:
%p a:= n-> numer(b(n)):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, May 12 2016
%t Table[Numerator@ SeriesCoefficient[Exp[x/(1 - x)], {x, 0, n}], {n, 19}] (* _Michael De Vlieger_, Dec 14 2015 *)
%t r[n_] := If[n == 0, 1, Hypergeometric1F1[1 - n, 2, -1]]; Table[Numerator@ r[n], {n, 0, 21}] (* _Peter Luschny_, Feb 02 2019 *)
%o (PARI) a(n) = numerator(sum(k=1, n, binomial(n-1, k-1)/k!)); \\ _Altug Alkan_, Dec 14 2015
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1-x)))); [Numerator(b[n]): n in [1..m]]; // _G. C. Greubel_, Dec 04 2018
%o (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
%Y Cf. A067653.
%K nonn,frac
%O 0,3
%A _Benoit Cloitre_, Feb 03 2002
%E a(0)=1 prepended by _Alois P. Heinz_, May 12 2016
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