Original entry on oeis.org
8, 240, 18264, 2845440, 792731520, 359935488000, 248962406889600
Offset: 3
A013587
Number of rational plane curves of degree d passing through 3d-1 general points.
Original entry on oeis.org
1, 1, 12, 620, 87304, 26312976, 14616808192, 13525751027392, 19385778269260800, 40739017561997799680, 120278021410937387514880, 482113680618029292368686080, 2551154673732472157928033617920, 17410560213476464590484763013222400
Offset: 1
Gary Kennedy (kennedy(AT)math.ohio-state.edu)
G.f. = x + x^2 + 12*x^3 + 620*x^4 + 87304*x^5 + 26312976*x^6 + ...
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- 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.
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- 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).
Cf.
A319851 (Welschinger invariants).
-
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);
-
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 *)
-
{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 */
-
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
A171118
Three-dimensional Welschinger invariants.
Original entry on oeis.org
1, -1, 45, -14589, 17756793, -58445425017, 426876362998821, -6061743911446054965, 152244625648721441783409
Offset: 0
- Aubin Arroyo, Erwan Brugallé and Lucía López de Medrano, Recursive formulas for Welschinger invariants of the projective plane, International Mathematics Research Notices, 2011, 1107-1134; arXiv:0809.1541 [math.AG], 2008-2010. See numbers W3(d) in Section 7.3.
- Erwan Brugallé and Penka Georgieva, Pencils of quadrics and Gromov-Witten-Welschinger invariants of CP^3, Mathematische Annalen 365, 363-380 (2016); arXiv:1508.02560 [math.AG], 2015-2016.
- Erwan Brugallé and Grigory Mikhalkin, Enumeration of curves via floor diagrams, C. R. Acad. Sci. Paris, Ser. I, 345 (2007), 329-334; arXiv:0706.0083 [math.AG], 2007.
Showing 1-3 of 3 results.