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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
 1, 1, 1, 1, 5, 1, 1, 16, 16, 1, 1, 42, 127, 42, 1, 1, 99, 715, 715, 99, 1, 1, 219, 3292, 7723, 3292, 219, 1, 1, 466, 13333, 63173, 63173, 13333, 466, 1, 1, 968, 49556, 429594, 861235, 429594, 49556, 968, 1, 1, 1981, 173570, 2567940, 9300303, 9300303, 2567940, 173570, 1981, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 3,5 COMMENTS 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 LINKS Tom Copeland, Combinatorics of OEIS-A074060 Posted Sept. 2008 Tom Copeland, Mathemagical Forests v2 Posted June 2008 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. M. Kontsevich and Y. Manin, Quantum cohomology of a product, arXiv:q-alg/9502009, 1995. Y. Manin, Generating functions in algebraic geometry and sums over trees, arXiv:alg-geom/9407005, 1994. - Tom Copeland, Dec 10 2011 M. A. Readdy, The pre-WDVV ring of physics and its topology, preprint, 2002. FORMULA 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 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 With the notation in Copeland's comments, dH(x,t)/dx = -g(H(x,t),t). - Tom Copeland, Sep 01 2011 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 EXAMPLE Viewed as a triangular array, the values are   1;   1,   1;   1,   5,   1;   1,  16,  16,   1;   1,  42, 127,  42,   1; ... MAPLE 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 MATHEMATICA DA = ((1+t) A[u, t] + u)/(1 - t A[u, t]); F = 0; Do[F = Integrate[Series[DA /. A[u, t] -> F, {u, 0, k}], u], {k, 1, 10}]; (cc = CoefficientList[#, t]; cc Denominator[cc[]])& /@ Drop[ CoefficientList[F, u], 2] // Flatten (* Jean-François Alcover, Oct 15 2019, after Eric Rains *) CROSSREFS Cf. A074059. 2nd diagonal is A002662. Sequence in context: A156920 A174044 A174159 * A157637 A157181 A029847 Adjacent sequences:  A074057 A074058 A074059 * A074061 A074062 A074063 KEYWORD nonn,tabl AUTHOR Margaret A. Readdy, Aug 16 2002 EXTENSIONS More terms from Eric Rains, Apr 02 2005 STATUS approved

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Last modified September 30 14:57 EDT 2020. Contains 337439 sequences. (Running on oeis4.)
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