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) ).
a(n) ~ c * d^n * n^(3*n-4), where d = 0.185519180960019376267112252210617741849455736227434091694584922574606814..., c = 8.73503626335165143920583748513754098083091109391517981485640427521559... - Vaclav Kotesovec, Apr 28 2024
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
KEYWORD
nonn,easy,nice
AUTHOR
Gary Kennedy (kennedy(AT)math.ohio-state.edu)
EXTENSIONS
Additional terms and references from Michael Somos
STATUS
approved