A013587 Number of rational plane curves of degree d passing through 3d-1 general points.

1, 1, 12, 620, 87304, 26312976, 14616808192, 13525751027392, 19385778269260800, 40739017561997799680, 120278021410937387514880, 482113680618029292368686080, 2551154673732472157928033617920, 17410560213476464590484763013222400

Offset: 1

Author

Gary Kennedy (kennedy(AT)math.ohio-state.edu)

Keywords

• nonn
• easy
• nice

Examples

			G.f. = x + x^2 + 12*x^3 + 620*x^4 + 87304*x^5 + 26312976*x^6 + ...
		

References

• M. Atiyah, On the unreasonable effectiveness of physics in mathematics, in "Highlights of Mathematical Physics", ed. A. S. Fokas, pp. 25ff.
• D. A. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, American Mathematical Society, 1999, p. 198.
• P. DiFranceso and C. Itzykson, Quantum intersection rings, in The Moduli Space of Curves, Birkhäuser, Boston, 1995, pp. 81-148.
• W. Fulton, Enumerative geometry via quantum cohomology, lecture notes, AMS Summer Institute, Santa Cruz, 1995.
• Yuri I. Manin, Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, American Mathematical Society, 1999, p. 7.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..169 (first 50 terms from T. D. Noe)
• Aubin Arroyo, Erwan Brugalle and Lucia Lopez de Medrano, Recursive formulas for Welschinger invariants of the projective plane, arXiv:0809.1541 [math.AG], 2008-2010. See 7.3 p. 16.
• Andrea Brini, Enumerative geometry of surfaces and topological strings, arXiv:2211.11037 [math-ph], 2022.
• Steven R. Finch, Enumerative geometry, February 24, 2014. [Cached copy, with permission of the author]
• Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, arXiv:0906.3828 [math.AG], 2009-2010. [From _N. J. A. Sloane_, Sep 27 2010]
• E. Getzler, Review of "Frobenius Manifolds, Quantum Cohomology and Moduli Spaces" by Y. I. Manin, Bull. Amer. Math. Soc., 38 (No. 1, 2001), 101-108.
• Étienne Ghys and Catriona Maclean, Des équations géométriques - Images des Mathématiques, CNRS, 2013.
• Daniel B. Grunberg and Pieter Moree, with an Appendix by Don Zagier, Sequences of enumerative geometry: congruences and asymptotics, arXiv:math/0610286 [math.NT], 2006.
• M. Kontsevich, Enumeration of rational curves via torus actions, in The Moduli Space of Curves, Birkhäuser, Boston, 1995, 335-368.
• M. Kontsevich, Enumeration of rational curves via torus actions, arXiv:hep-th/9405035, 1994-1995.
• Grigory Mikhalkin, Enumerative tropical algebraic geometry in R^2, arXiv:math/0312530 [math.AG], 2003-2004. [From _N. J. A. Sloane_, Sep 27 2010]
• Alexis Roquefeuil, Confluence of quantum K-theory to quantum cohomology for projective spaces, arXiv:1911.00254 [math.AG], 2019.
• Ian Strachan, How to count curves: from C. 19 problems to C. 20 solutions, Phil. Trans. Royal Soc. London, A 351 (2003), 2633-2647.
• Jean-Yves Welschinger, Enumération de fractions rationnelles réelles, Images des Mathématiques, CNRS, 2006 (in French).

Crossrefs

Cf. A319851 (Welschinger invariants).

Programs

Maple

a:= proc(d::nonnegint) option remember; if d = 1 then 1 else
       add(a(k)*a(d-k)*(k^2*(d-k)^2*binomial(3*d-4, 3*k-2)-k^3*(d-k)
       *binomial(3*d-4, 3*k-1)), k = 1 .. d-1) fi
    end:
seq(a(n), n=1..20);

Mathematica

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 *)

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 */

PARI

N=20;
MEM=Vec(1, N);  \\ for memoization
K(d)= \\ Kontsevich's recursion, see S. Finch link.
{
    my(m = MEM[d]);
    if ( m, return(m) );  \\ memoized
    MEM[d] =              \\ memoize and return
       sum(d1=1, d-1, my( d2 = d-d1 ); \\ d1+d2=d, both >= 1 and < d
          K(d1) * K(d2) *
          (d1^2 * d2^2 * binomial(3*d-4, 3*d1-2) -
           d1^3 * d2^1 * binomial(3*d-4, 3*d1-1) )
    );
}
vector(N, d, K(d) )
\\ Joerg Arndt, Feb 26 2014; minor edits by M. F. Hasler, Jul 25 2025

Formula

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) ).

a(n) ~ c * d^n * n^(3*n-4), where d = 0.185519180960019376267112252210617741849455736227434091694584922574606814..., c = 8.73503626335165143920583748513754098083091109391517981485640427521559... - Vaclav Kotesovec, Apr 28 2024

Extensions

Additional terms and references from Michael Somos