login
Denominators of power series arising in random graphs.
1

%I #12 Aug 13 2018 09:07:28

%S 1,3,9,135,405,189,42525,18225,229635,189448875,795685275,14105329875,

%T 9499507875,107417512125,142492618125,4154372281434375,

%U 12463116844303125,125364292963284375,50235263108858953125,7931883648767203125,80559094658206539375,218372688734209869234375,5419613093130844936453125,20735910965022363235125

%N Denominators of power series arising in random graphs.

%C Numerators are A141142. 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, 24}]] //

%t CoefficientList[#, z]& // Denominator (* _Jean-François Alcover_, Aug 13 2018, after _Max Alekseyev_ *)

%Y Cf. A141142.

%K frac,nonn

%O 1,2

%A _Jonathan Vos Post_, Jun 10 2008

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