A072597 Expansion of 1/(exp(-x) - x) as exponential generating function.

1, 2, 7, 37, 261, 2301, 24343, 300455, 4238153, 67255273, 1185860331, 23000296155, 486655768525, 11155073325917, 275364320099807, 7282929854486431, 205462851526617489, 6158705454187353297, 195465061563672788947, 6548320737474275229347, 230922973019493881984021

Offset: 0

Author

Michael Somos, Jun 23 2002

Keywords

• nonn
• easy

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? - 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

Call an element g in a semigroup a group element if g^j = g for some j > 1. Then a(n) is the number of group elements in the semigroup of partial transformations of an n-set. Hence a(n) = Sum_{k=0..n} A154372(n,k)*k!. - Geoffrey Critzer, Nov 27 2021

Examples

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

References

• O. Ganyushkin and V Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 70.
• 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 [cs.FL], 2012. - From _N. J. A. Sloane_, Sep 21 2012

Crossrefs

Cf. A000110, A006153, A030178, A089148.

Cf. A048993, A052820.

Cf. A006153, A154372.

Programs

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 *)

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 */

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

a(n) = A006153(n+1)/(n+1). - Seiichi Manyama, Nov 05 2024