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).
1, 1, 2, 7, 34, 213, 1630, 14747, 153946, 1821473, 24087590, 352080111, 5636451794, 98081813581, 1843315388078, 37209072076483, 802906142007946, 18443166021077145, 449326835001457846, 11572432709175470807, 314160322966817351938, 8965995574654847062469
Offset: 1
Keywords
Examples
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)
Links
- Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
- V. M. Buchstaber and A. P. Veselov, Differential algebra of polytopes and inversion formulas, arXiv:2402.07168 [math.CO], 2024. See p. 9.
- 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.
- Curtis T. McMullen, Moduli spaces in genus zero and inversion of power series, (2012).
- 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.
- Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.
Crossrefs
Row sums of triangle A074060.
Programs
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Maple
series(exp(LambertW(-exp(-2)*(2+x))+2)-1,x,30): A:=simplify(%,symbolic): A074059:=n->n!*coeff(A,x,n): # Gessel
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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-François Alcover, Jun 20 2013 *) -
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
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
a(1) = 1; a(n) = a(n-1) + Sum_{k=2..n-1} binomial(n-1,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Aug 28 2020
Extensions
More terms from Ira M. Gessel, Dec 15 2005
a(20)-a(22) from Stefano Spezia, Feb 14 2024