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%I #79 Oct 19 2024 15:57:32
%S 1,1,12,620,87304,26312976,14616808192,13525751027392,
%T 19385778269260800,40739017561997799680,120278021410937387514880,
%U 482113680618029292368686080,2551154673732472157928033617920,17410560213476464590484763013222400
%N Number of rational plane curves of degree d passing through 3d-1 general points.
%D M. Atiyah, On the unreasonable effectiveness of physics in mathematics, in "Highlights of Mathematical Physics", ed. A. S. Fokas, pp. 25ff.
%D D. A. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, American Mathematical Society, 1999, p. 198.
%D P. DiFranceso and C. Itzykson, Quantum intersection rings, in The Moduli Space of Curves, Birkhäuser, Boston, 1995, pp. 81-148.
%D W. Fulton, Enumerative geometry via quantum cohomology, lecture notes, AMS Summer Institute, Santa Cruz, 1995.
%D Yuri I. Manin, Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, American Mathematical Society, 1999, p. 7.
%H Alois P. Heinz, <a href="/A013587/b013587.txt">Table of n, a(n) for n = 1..169</a> (first 50 terms from T. D. Noe)
%H Aubin Arroyo, Erwan Brugalle and Lucia Lopez de Medrano, <a href="https://arxiv.org/abs/0809.1541">Recursive formulas for Welschinger invariants of the projective plane</a>, arXiv:0809.1541 [math.AG], 2008-2010. See 7.3 p. 16.
%H Andrea Brini, <a href="https://arxiv.org/abs/2211.11037">Enumerative geometry of surfaces and topological strings</a>, arXiv:2211.11037 [math-ph], 2022.
%H Steven R. Finch, <a href="/A013587/a013587.pdf">Enumerative geometry</a>, February 24, 2014. [Cached copy, with permission of the author]
%H Sergey Fomin and Grigory Mikhalkin, <a href="http://arxiv.org/abs/0906.3828">Labeled floor diagrams for plane curves</a>, arXiv:0906.3828 [math.AG], 2009-2010. [From _N. J. A. Sloane_, Sep 27 2010]
%H E. Getzler, <a href="http://www.ams.org/journal-getitem?pii=S0273-0979-00-00888-0">Review of "Frobenius Manifolds, Quantum Cohomology and Moduli Spaces"</a> by Y. I. Manin, Bull. Amer. Math. Soc., 38 (No. 1, 2001), 101-108.
%H Étienne Ghys and Catriona Maclean, <a href="http://images-archive.math.cnrs.fr/Des-equations-geometriques.html">Des équations géométriques</a> - Images des Mathématiques, CNRS, 2013.
%H Daniel B. Grunberg and Pieter Moree, with an Appendix by Don Zagier, <a href="http://arxiv.org/abs/math.NT/0610286">Sequences of enumerative geometry: congruences and asymptotics</a>, arXiv:math/0610286 [math.NT], 2006.
%H M. Kontsevich, <a href="http://dx.doi.org/10.1007/978-1-4612-4264-2_12">Enumeration of rational curves via torus actions</a>, in The Moduli Space of Curves, Birkhäuser, Boston, 1995, 335-368.
%H M. Kontsevich, <a href="https://arxiv.org/abs/hep-th/9405035">Enumeration of rational curves via torus actions</a>, arXiv:hep-th/9405035, 1994-1995.
%H Grigory Mikhalkin, <a href="http://arxiv.org/abs/math/0312530">Enumerative tropical algebraic geometry in R^2</a>, arXiv:math/0312530 [math.AG], 2003-2004. [From _N. J. A. Sloane_, Sep 27 2010]
%H Alexis Roquefeuil, <a href="https://arxiv.org/abs/1911.00254">Confluence of quantum K-theory to quantum cohomology for projective spaces</a>, arXiv:1911.00254 [math.AG], 2019.
%H Ian Strachan, <a href="http://www.maths.gla.ac.uk/~iabs/Visions.pdf">How to count curves: from C. 19 problems to C. 20 solutions</a>, Phil. Trans. Royal Soc. London, A 351 (2003), 2633-2647.
%H Jean-Yves Welschinger, <a href="http://images-archive.math.cnrs.fr/Enumeration-de-fractions.html">Enumération de fractions rationnelles réelles</a>, Images des Mathématiques, CNRS, 2006 (in French).
%F a_d = Sum_{i+j=d} a_i*a_j ( i^2*j^2*binomial(3d-4, 3i-2) - i^3*j*binomial(3d-4, 3i-1) ).
%F a(n) ~ c * d^n * n^(3*n-4), where d = 0.185519180960019376267112252210617741849455736227434091694584922574606814..., c = 8.73503626335165143920583748513754098083091109391517981485640427521559... - _Vaclav Kotesovec_, Apr 28 2024
%e G.f. = x + x^2 + 12*x^3 + 620*x^4 + 87304*x^5 + 26312976*x^6 + ...
%p a:= proc(d::nonnegint) option remember; if d = 1 then 1 else
%p add(a(k)*a(d-k)*(k^2*(d-k)^2*binomial(3*d-4, 3*k-2)-k^3*(d-k)
%p *binomial(3*d-4, 3*k-1)), k = 1 .. d-1) fi
%p end:
%p seq(a(n), n=1..20);
%t a[n_] := a[n] = Sum[ a[k]*a[n-k]*k^2*(n-k)*(3k-n)*(3n-4)! / (3k-1)! / (3*(n-k)-2)!, {k, 1, n-1}]; a[1] = 1; Table[a[n], {n, 1, 13}] (* _Jean-François Alcover_, Nov 09 2011, after PARI *)
%o (PARI) {a(n) = if( n<2, n>0, sum(k=1, n-1, a(k) * a(n-k) * k^2 * (n-k) * (3*k-n) * (3*n-4)! / ((3*k-1)! * (3*(n-k)-2)!) ))}; /* _Michael Somos_, Dec 11 1999 */
%o (PARI)
%o N=20;
%o MEM=vector(N, j, -1); \\ for memoization
%o MEM[1] = 1;
%o K(d)= \\ Kontsevich's recursion, see S. Finch link.
%o {
%o my(m = MEM[d]);
%o if ( m != -1, return(m) ); \\ memoized
%o my(t, d2);
%o t = sum(d1=1, d-1, d2=d-d1; \\ d1+d2==d, both >= 1
%o K(d1) * K(d2) *
%o (d1^2 * d2^2 * binomial(3*d-4, 3*d1-2) -
%o d1^3 * d2^1 * binomial(3*d-4, 3*d1-1) )
%o );
%o MEM[d] = t; \\ memoize
%o return(t);
%o }
%o vector(N, d, K(d) )
%o \\ _Joerg Arndt_, Feb 26 2014
%K nonn,easy,nice
%O 1,3
%A Gary Kennedy (kennedy(AT)math.ohio-state.edu)
%E Additional terms and references from _Michael Somos_