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A067764 Numerators of the coefficients in exp(x/(1-x)) power series. 7
1, 1, 3, 13, 73, 167, 4051, 37633, 43817, 4596553, 58941091, 274691047, 12470162233, 202976401213, 1178339174801, 65573803186921, 99264170666917, 994319127823939, 588633468315403843, 13564373693588558173, 109232642628695218147, 752832094524169066031 (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

The ratio sequence given by c(n) = A067764(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

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..445

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 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

EXAMPLE

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! + ... = 73e/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 + ... .

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 *)

PROG

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

CROSSREFS

Cf. A067653.

Sequence in context: A086662 A293195 A090754 * A193930 A063512 A199317

Adjacent sequences:  A067761 A067762 A067763 * A067765 A067766 A067767

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 October 23 06:13 EDT 2018. Contains 316519 sequences. (Running on oeis4.)