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

1,3

Links

Table of n, a(n) for n=1..22.

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.

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

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

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-François 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: A249833 A111539 A337000 * 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

a(20)-a(22) from Stefano Spezia, Feb 14 2024

Status

approved