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A072597 Expansion of 1/(exp(-x)-x) as exponential generating function. 6
1, 2, 7, 37, 261, 2301, 24343, 300455, 4238153, 67255273, 1185860331, 23000296155, 486655768525, 11155073325917, 275364320099807, 7282929854486431, 205462851526617489, 6158705454187353297 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Polynomials from A140749/A141412 are linked to Stirling1 (see A048594, A129841, A140749).See also P. Flajolet, X. Gourdon, B. Salvy in, available on Internet, RR-1857.pdf (preprint of unavailable Gazette des Mathematiciens 55, 1993, pp.67-78.For graph 2 see also X. Gourdon RR-1852.pdf, pp.64-65). What is the corresponding graph for A152650/A152656 =simplified A009998/A119502 linked, via A152818, to a(n), then Stirling2? [From Paul Curtz, Dec 16 2008]

Denominators in rational approximations of Lambert W(1). See Ramanujan, Notebooks, volume 2, page 22: "2. If e^{-x} = x, shew that the convergents to x are 1/2, 4/7, 21/37, 148/261, &c." Numerators in A006153. - Michael Somos, Jan 21 2019

REFERENCES

S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 22.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..411

W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013.

G. Jiraskova and J. Shallit, The state complexity of star-complement-star, arXiv preprint arXiv:1203.5353, 2012. - From N. J. A. Sloane, Sep 21 2012

FORMULA

E.g.f.: 1 / (exp(-x) - x).

a(n) = n!*Sum_{k=0..n} (n-k+1)^k/k!. - Vladeta Jovovic, Aug 31 2003

a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*A052820(k). - Vladeta Jovovic, Apr 12 2004

Recurrence : a(n+1) = 1 + sum { j=1, n, binomial(n, j)*a(j)*j } - Jon Perry, Apr 25 2005

E.g.f.: 1/(Q(0) - x) where Q(k) = 1 - x/(2*k+1 - x*(2*k+1)/(x - (2*k+2)/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 04 2013

a(n) ~ n!/((1+c)*c^(n+1)), where c = A030178 = LambertW(1) = 0.5671432904... - Vaclav Kotesovec, Jun 26 2013

O.g.f.: Sum_{k>=0} k!*x^k/(1 - (k + 1)*x)^(k+1). - Ilya Gutkovskiy, Oct 09 2018

EXAMPLE

G.f. = 1 + 2*x + 7*x^2 + 37*x^3 + 261*x^4 + 2301*x^5 + 24343*x^6 + ...

MATHEMATICA

CoefficientList[Series[1/(Exp[-x]-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / (Exp[-x] - x), {x, 0, n}]]; (* Michael Somos, Jan 21 2019 *)

a[ n_] := If[ n < 0, 0, n! Sum[ (n - k + 1)^k / k!, {k, 0, n}]]; (* Michael Somos, Jan 21 2019 *)

PROG

(PARI) {a(n) = if( n<0, 0, n! * polcoeff( 1 / (exp(-x + x * O(x^n)) - x), n))};

(PARI) {a(n) = if( n<0, 0, n! * sum(k=0, n, (n-k+1)^k / k!))}; /* Michael Somos, Jan 21 2019 */

CROSSREFS

Cf. A000110, A006153, A089148.

Sequence in context: A302859 A135164 A321087 * A322140 A125515 A135920

Adjacent sequences:  A072594 A072595 A072596 * A072598 A072599 A072600

KEYWORD

nonn,easy

AUTHOR

Michael Somos, Jun 23 2002

STATUS

approved

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Last modified September 17 10:52 EDT 2019. Contains 327129 sequences. (Running on oeis4.)