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%I #52 Feb 14 2024 14:26:54
%S 1,1,2,7,34,213,1630,14747,153946,1821473,24087590,352080111,
%T 5636451794,98081813581,1843315388078,37209072076483,802906142007946,
%U 18443166021077145,449326835001457846,11572432709175470807,314160322966817351938,8965995574654847062469
%N 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).
%H Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%H V. M. Buchstaber and A. P. Veselov, <a href="https://arxiv.org/abs/2402.07168">Differential algebra of polytopes and inversion formulas</a>, arXiv:2402.07168 [math.CO], 2024. See p. 9.
%H Brian Drake, Ira M. Gessel, and Guoce Xin, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Gessel/gessel20.html">Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry,</a> J. of Integer Sequences, Vol. 10 (2007), #07.3.7.
%H I. P. Goulden, S. Litsyn, and V. Shevelev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Litsyn/litsyn16.html">On a Sequence Arising in Algebraic Geometry</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.7.
%H S. Keel, <a href="http://dx.doi.org/10.1090/S0002-9947-1992-1034665-0">Intersection theory of moduli space of stable n-pointed curves of genus zero</a>, Trans. Amer. Math. Soc. 330 (1992), 545-574.
%H M. Kontsevich and Y. Manin, <a href="http://dx.doi.org/10.1007/s002220050055">Quantum cohomology of a product</a>, (with Appendix by R. Kaufmann), Inv. Math. 124, f. 1-3 (1996) 313-339.
%H Margaret Readdy, <a href="http://arxiv.org/abs/math/0511420">The pre-WDVV ring of physics and its topology</a>, arXiv:math/0511420 [math.CO], The Ramanujan Journal, Special issue on the Number Theory and Combinatorics in Physics, 10 (2005), 269-281.
%F 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
%F E.g.f.: Series_Reversion[ x - Sum_{n>=2} (-x)^n/(n(n-1)) ]. - _Paul D. Hanna_, Sep 24 2010
%F 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
%F 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
%F 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
%F 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
%F a(n) ~ n^(n-1) / (exp(1)*(exp(1)-2))^(n-1/2). - _Vaclav Kotesovec_, Oct 05 2013
%F 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
%e From _Paul D. Hanna_, Sep 24 2010: (Start)
%e E.g.f.: x + x^2/2! + 2*x^3/3! + 7*x^4/4! + 34*x^5/5! + 213*x^6/6! +...
%e The series reversion of the e.g.f. begins:
%e x - x^2/2 + x^3/6 - x^4/12 + x^5/20 - x^6/30 + x^7/42 - x^8/56 +... (End)
%p series(exp(LambertW(-exp(-2)*(2+x))+2)-1,x,30): A:=simplify(%,symbolic): A074059:=n->n!*coeff(A,x,n): # Gessel
%t 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 *)
%o (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
%Y Cf. A074060.
%K nonn
%O 1,3
%A _Margaret A. Readdy_, Aug 16 2002
%E More terms from _Ira M. Gessel_, Dec 15 2005
%E a(20)-a(22) from _Stefano Spezia_, Feb 14 2024
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