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Search: id:A074060
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| A074060 |
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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). |
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+0
5
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| 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; listen)
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OFFSET
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3,5
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COMMENT
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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. [From Tom Copeland (tcjpn(AT)msn.com), Sep 28 2008]
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REFERENCES
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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. A. Readdy, The pre-WDVV ring of physics and its topology, preprint, 2002.
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LINKS
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Tom Copeland, Combinatorics of OEIS-A074060 Posted Sept. 2008 [From Tom Copeland (tcjpn(AT)msn.com), Sep 28 2008]
Tom Copeland, Mathemagical Forests v2 Posted June 2008 [From Tom Copeland (tcjpn(AT)msn.com), Sep 28 2008]
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FORMULA
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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+2x)^(1/2), where W is a branch of the Lambert function such that W[ -2*e^(-2)] = -2 . - Tom Copeland (tcjpn(AT)msn.com), 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 (tcjpn(AT)msn.com), Jun 01 2008
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EXAMPLE
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Viewed as a triangular array, the values are: 1; 1 1; 1 5 1; 1 16 16 1; 1 42 127 42 1; ...
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MAPLE
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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:
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CROSSREFS
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Cf. A074059. 2nd diagonal is A002662.
Sequence in context: A141691 A157147 A156920 this_sequence A157637 A157181 A029847
Adjacent sequences: A074057 A074058 A074059 this_sequence A074061 A074062 A074063
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KEYWORD
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nonn,tabl
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AUTHOR
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Margaret A. Readdy (readdy(AT)ms.uky.edu), Aug 16 2002
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EXTENSIONS
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More terms and Maple code from Eric Rains, Apr 02 2005
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