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A074060 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). 5

%I

%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-restricted 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

%H Tom Copeland, <a href="http://tcjpn.spaces.live.com/default.aspx">Combinatorics of OEIS-A074060</a> Posted Sept. 2008

%H Tom Copeland, <a href="http://tcjpn.spaces.live.com/default.aspx">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

%H Y. Manin, <a href="http://arxiv.org/abs/alg-geom/9407005">Generating functions in algebraic geometry and sums over trees</a> - from _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

%Y Cf. A074059. 2nd diagonal is A002662.

%K nonn,tabl,changed

%O 3,5

%A Margaret A. Readdy (readdy(AT)ms.uky.edu), Aug 16 2002

%E More terms from Eric Rains, Apr 02 2005

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