A141143 Denominators of power series arising in random graphs.

1, 3, 9, 135, 405, 189, 42525, 18225, 229635, 189448875, 795685275, 14105329875, 9499507875, 107417512125, 142492618125, 4154372281434375, 12463116844303125, 125364292963284375, 50235263108858953125, 7931883648767203125, 80559094658206539375, 218372688734209869234375, 5419613093130844936453125, 20735910965022363235125

Offset

1,2

Comments

Numerators are A141142. Series given in Bouchard and Marino 2.31, p. 5, with citations to earlier literature.

Links

Table of n, a(n) for n=1..24.

Vincent Bouchard and Marcos Marino, Hurwitz numbers, matrix models and enumerative geometry, arXiv:0709.1458 [math.AG], 2007-2008.

Formula

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

s(z) = -1 - LambertW(-(1+z)*exp(-z-1)). - Max Alekseyev, Aug 17 2013

Example

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).

Maple

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

Mathematica

Assuming[z>0, Series[-1 - ProductLog[-(1+z) Exp[-1-z]], {z, 0, 24}]] //
CoefficientList[#, z]& // Denominator (* Jean-François Alcover, Aug 13 2018, after Max Alekseyev *)

Crossrefs

Cf. A141142.

Sequence in context: A062702 A289169 A257032 * A368092 A163402 A288757

Adjacent sequences: A141140 A141141 A141142 * A141144 A141145 A141146

Keyword

frac,nonn

Author

Jonathan Vos Post, Jun 10 2008

Extensions

Terms a(8) onward from Max Alekseyev, Aug 17 2013

Status

approved