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A074059 Dimension of the cohomology ring of the moduli space of n-pointed curves of genus 0 satisfying the associativity equations of physics (also known as the WDVV equations). 6
1, 1, 2, 7, 34, 213, 1630, 14747, 153946, 1821473, 24087590, 352080111, 5636451794, 98081813581, 1843315388078, 37209072076483, 802906142007946, 18443166021077145, 449326835001457846 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Table of n, a(n) for n=1..19.

Brian Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), #07.3.7.

I. P. Goulden, S. Litsyn and V. Shevelev, On a Sequence Arising in Algebraic Geometry, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.7.

S. Keel, Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545-574.

M. Kontsevich and Y. Manin, Quantum cohomology of a product, (with Appendix by R. Kaufmann), Inv. Math. 124, f. 1-3 (1996) 313-339.

Margaret Readdy, The pre-WDVV ring of physics and its topology, arXiv:math/0511420 [math.CO], The Ramanujan Journal, Special issue on the Number Theory and Combinatorics in Physics, 10 (2005), 269-281.

FORMULA

The exponential generating function A = A(x) = sum_{n>=1} a(n) x^n/n! satisfies the equation (1+A)log(1+A) = 2A-x. Explicitly, 1+A(x) = exp(2+W(e^(-2)(2+x))), where W is Lambert's W-function. - Ira M. Gessel, Dec 15 2005

E.g.f.: Series_Reversion[ x - Sum_{n>=2} (-x)^n/(n(n-1)) ]. - Paul D. Hanna, Sep 24 2010

Let h(x) = 1/(1-log(1+x)), then a(n) = ((h(x)*d/dx)^n)x evaluated at x=0, i.e., A(x) = exp(x*a(.)) = exp(x*h(u)*d/du) u, evaluated at u=0. Also, dA(x)/dx = h(A(x)). - Tom Copeland, Sep 06 2011

An o.g.f. is provided by the integral from w=0 to infinity of exp(-2w) * (1+z*w)^((1+z*w)/z). - Tom Copeland, Sep 09 2011

E.g.f. = -1/{1+W[-(2+x) exp(-2)]} with W(x) the Monir branch of the Lambert W fct. defined in A135338 and offset 0. - Tom Copeland, Oct 05 2011

a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator 1/(1-x)*exp(-x)*d/dx. Cf. A061356. - Peter Bala, Dec 08 2011

a(n) ~ n^(n-1) / (exp(1)*(exp(1)-2))^(n-1/2). - Vaclav Kotesovec, Oct 05 2013

EXAMPLE

From Paul D. Hanna, Sep 24 2010: (Start)

E.g.f.: x + x^2/2! + 2*x^3/3! + 7*x^4/4! + 34*x^5/5! + 213*x^6/6! +...

The series reversion of the e.g.f. begins:

x - x^2/2 + x^3/6 - x^4/12 + x^5/20 - x^6/30 + x^7/42 - x^8/56 +... (End)

MAPLE

series(exp(LambertW(-exp(-2)*(2+x))+2)-1, x, 30): A:=simplify(%, symbolic): A074059:=n->n!*coeff(A, x, n): (Gessel)

MATHEMATICA

max = 19; $Assumptions = x > 0; (Series[ Exp[2 + ProductLog[-1, -(x+2)/E^2]] - 1, {x, 0, 19}] // CoefficientList[#, x] &) * Range[0, 19]! // Rest (* Jean-Fran├žois Alcover, Jun 20 2013 *)

PROG

(PARI) {a(n)=if(n<1, 0, n!*polcoeff(serreverse(x-sum(k=2, n, (-x)^k/(k*(k-1)))+x*O(x^n)), n))} \\ Paul D. Hanna, Sep 24 2010

CROSSREFS

Cf. A074060.

Sequence in context: A234239 A249833 A111539 * A177401 A171792 A185324

Adjacent sequences:  A074056 A074057 A074058 * A074060 A074061 A074062

KEYWORD

nonn

AUTHOR

Margaret A. Readdy, Aug 16 2002

EXTENSIONS

More terms from Ira M. Gessel, Dec 15 2005

STATUS

approved

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Last modified April 25 08:35 EDT 2018. Contains 303055 sequences. (Running on oeis4.)