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

1,3

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).

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) ).

Example

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

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 *)

Prog

(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=vector(N, j, -1);  \\ for memoization
MEM[1] = 1;
K(d)= \\ Kontsevich's recursion, see S. Finch link.
{
    my(m = MEM[d]);
    if ( m != -1, return(m) );  \\ memoized
    my(t, d2);
    t = sum(d1=1, d-1,  d2=d-d1;  \\ d1+d2==d, both >= 1
        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) )
    );
    MEM[d] = t;  \\ memoize
    return(t);
}
vector(N, d, K(d) )
\\ Joerg Arndt, Feb 26 2014

Crossrefs

Sequence in context: A220368 A220312 A288686 * A126159 A220992 A241227

Adjacent sequences: A013584 A013585 A013586 * A013588 A013589 A013590

Keyword

nonn,easy,nice

Author

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

Extensions

Additional terms and references from Michael Somos

Status

approved