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%I #58 Jun 27 2021 16:23:12
%S 1,1,1,1,5,1,1,16,16,1,1,42,127,42,1,1,99,715,715,99,1,1,219,3292,
%T 7723,3292,219,1,1,466,13333,63173,63173,13333,466,1,1,968,49556,
%U 429594,861235,429594,49556,968,1,1,1981,173570,2567940,9300303,9300303,2567940,173570,1981,1
%N Graded 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).
%C Combinatorial interpretations of Lagrange inversion (A134685) and the 2-Stirling numbers of the first kind (A049444 and A143491) provide a combinatorial construction for A074060 (see first Copeland link). For relations of A074060 to other arrays see second Copeland link page 19. - _Tom Copeland_, Sep 28 2008
%C These Poincare polynomials for the compactified moduli space of rational curves are presented on p. 5 of Lando and Zvonkin as well as those for the non-compactified Poincare polynomials of A049444 in factorial form. - _Tom Copeland_, Jun 13 2021
%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2008/09/28/combinatorics-of-oeis-a074060/">Combinatorics of OEIS-A074060</a>, Posted Sept. 2008.
%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2008/06/12/mathemagical-forests/">Mathemagical Forests v2</a>, Posted June 2008.
%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 M. Kontsevich and Y. Manin, <a href="http://arxiv.org/abs/q-alg/9502009">Quantum cohomology of a product</a>, arXiv:q-alg/9502009, 1995.
%H S. Lando and A. Zvonkin, <a href="http://dx.doi.org/10.1007/978-3-540-38361-1">Graphs on surfaces and their applications</a>, Encyclopaedia of Mathematical Sciences, 141, Springer, 2004.
%H Y. Manin, <a href="http://arxiv.org/abs/alg-geom/9407005">Generating functions in algebraic geometry and sums over trees</a>, arXiv:alg-geom/9407005, 1994. - _Tom Copeland_, Dec 10 2011
%H M. A. Readdy, <a href="http://www.ms.uky.edu/~readdy/Papers/pre_WDVV.pdf">The pre-WDVV ring of physics and its topology</a>, preprint, 2002.
%F Define offset to be 0 and P(n,t) = (-1)^n Sum_{j=0..n-2} a(n-2,j)*t^j with P(1,t) = -1 and P(0,t) = 1, then H(x,t) = -1 + exp(P(.,t)*x) is the compositional inverse in x about 0 of G(x,t) in A049444. H(x,0) = exp(-x) - 1, H(x,1) = -1 + exp( 2 + W( -exp(-2) * (2-x) ) ) and H(x,2) = 1 - (1+2*x)^(1/2), where W is a branch of the Lambert function such that W(-2*exp(-2)) = -2. - _Tom Copeland_, Feb 17 2008
%F Let offset=0 and g(x,t) = (1-t)/((1+x)^(t-1)-t), then the n-th row polynomial of the table is given by [(g(x,t)*D_x)^(n+1)]x with the derivative evaluated at x=0. - _Tom Copeland_, Jun 01 2008
%F With the notation in Copeland's comments, dH(x,t)/dx = -g(H(x,t),t). - _Tom Copeland_, Sep 01 2011
%F The term linear in x of [x*g(d/dx,t)]^n 1 gives the n-th row polynomial with offset 1. (See A134685.) - _Tom Copeland_, Oct 21 2011
%e Viewed as a triangular array, the values are
%e 1;
%e 1, 1;
%e 1, 5, 1;
%e 1, 16, 16, 1;
%e 1, 42, 127, 42, 1; ...
%p DA:=((1+t)*A(u,t)+u)/(1-t*A(u,t)): F:=0: for k from 1 to 10 do F:=map(simplify,int(series(subs(A(u,t)=F,DA),u,k),u)); od: # Eric Rains, Apr 02 2005
%t DA = ((1+t) A[u, t] + u)/(1 - t A[u, t]); F = 0;
%t Do[F = Integrate[Series[DA /. A[u, t] -> F, {u, 0, k}], u], {k, 1, 10}];
%t (cc = CoefficientList[#, t]; cc Denominator[cc[[1]]])& /@ Drop[ CoefficientList[F, u], 2] // Flatten (* _Jean-François Alcover_, Oct 15 2019, after Eric Rains *)
%Y Cf. A074059. 2nd diagonal is A002662.
%K nonn,tabl
%O 3,5
%A _Margaret A. Readdy_, Aug 16 2002
%E More terms from Eric Rains, Apr 02 2005
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