A072597 - OEIS

%I #64 Feb 03 2022 05:06:00

%S 1,2,7,37,261,2301,24343,300455,4238153,67255273,1185860331,

%T 23000296155,486655768525,11155073325917,275364320099807,

%U 7282929854486431,205462851526617489,6158705454187353297,195465061563672788947,6548320737474275229347,230922973019493881984021

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

%C 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

%C 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

%C 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

%D O. Ganyushkin and V Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 70.

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

%H Seiichi Manyama, <a href="/A072597/b072597.txt">Table of n, a(n) for n = 0..411</a>

%H W. S. Gray and M. Thitsa, <a href="http://dx.doi.org/10.1109/SSST.2013.6524939">System Interconnections and Combinatorial Integer Sequences</a>, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013.

%H G. Jiraskova and J. Shallit, <a href="http://arxiv.org/abs/1203.5353">The state complexity of star-complement-star</a>, arXiv preprint arXiv:1203.5353 [cs.FL], 2012. - From _N. J. A. Sloane_, Sep 21 2012

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

%F a(n) = n!*Sum_{k=0..n} (n-k+1)^k/k!. - _Vladeta Jovovic_, Aug 31 2003

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*A052820(k). - _Vladeta Jovovic_, Apr 12 2004

%F Recurrence: a(n+1) = 1 + Sum_{j=1..n} binomial(n, j)*a(j)*j. - _Jon Perry_, Apr 25 2005

%F 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

%F a(n) ~ n!/((1+c)*c^(n+1)), where c = A030178 = LambertW(1) = 0.5671432904... - _Vaclav Kotesovec_, Jun 26 2013

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

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

%t CoefficientList[Series[1/(Exp[-x]-x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 26 2013 *)

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

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

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

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

%Y Cf. A000110, A006153, A030178, A089148.

%Y Cf. A048993, A052820.

%Y Cf. A154372.

%K nonn,easy

%O 0,2

%A _Michael Somos_, Jun 23 2002