A141142 - OEIS

%I #12 Aug 13 2018 09:05:01

%S -1,2,-4,44,-104,40,-7648,2848,-31712,23429344,-89072576,1441952704,

%T -893393408,9352282112,-11547336704,314833934543872,-886909037097472,

%U 8407858707080704,-3185585650598165504,476968653230369792,-4605749416183789568,11898401315301146359808,-282034907680992595910656,1032668724971184398336,-2345904699036953797820416

%N Numerators of power series arising in random graphs.

%C Denominators are A141143. Series given in Bouchard and Marino 2.31, p. 5, with citations to earlier literature.

%H Vincent Bouchard and Marcos Marino, <a href="https://arxiv.org/abs/0709.1458">Hurwitz numbers, matrix models and enumerative geometry</a>, arXiv:0709.1458 [math.AG], 2007-2008.

%F Power series s(z) defined by (1+z)*exp(-z) = (1 + s(z))*exp(-s(z)).

%F s(z) = -1 - LambertW(-(1+z)*exp(-z-1)). - _Max Alekseyev_, Aug 17 2013

%e s(z) = -z + (2/3)*z^2 - (4/9)*z^3 + (44/135)*z^4 - (104/405)*z^5 + (40/189)*z^6 - (7648/42525)*z^7 + O(z^8).

%p assume(z>0); series( -1 - LambertW(-(1+z)*exp(-z-1)), z, 20 );

%t Assuming[z>0, Series[-1 - ProductLog[-(1+z) Exp[-1-z]], {z, 0, 25}]] // CoefficientList[#, z]& // Numerator (* _Jean-François Alcover_, Aug 13 2018, after _Max Alekseyev_ *)

%Y Cf. A141143.

%K frac,sign

%O 1,2

%A _Jonathan Vos Post_, Jun 10 2008

%E Terms a(8) onward from _Max Alekseyev_, Aug 17 2013