login
This site is supported by donations to The OEIS Foundation.
Logo

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A013587 Number of rational plane curves of degree d passing through 3d-1 general points. 1
1, 1, 12, 620, 87304, 26312976, 14616808192, 13525751027392, 19385778269260800, 40739017561997799680, 120278021410937387514880, 482113680618029292368686080, 2551154673732472157928033617920 (list; graph; refs; listen; history; internal format)
OFFSET

1,3

REFERENCES

M. Atiyah, On the unreasonable effectiveness of physics in mathematics, in "Highlights of Mathematical Physics:, ed. A. S. Fokas, pp. 25-.

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, Birkhaeuser, Boston, 1995, pp. 81-148.

Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, arXiv:0906.3828. [From N. J. A. Sloane, Sep 27 2010]

W. Fulton, Enumerative geometry via quantum cohomology, lecture notes, AMS Summer Institute, Santa Cruz, 1995.

Daniel B. Grunberg and Pieter Moree, with an Appendix by Don Zagier, Sequences of enumerative geometry: congruences and asymptotics, arXiv math.NT/0610286.

M. Kontsevich, Enumeration of rational curves via torus actions, in The Moduli Space of Curves, Birkhaeuser, Boston, 1995, 335-368.

Yuri I. Manin, Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, American Mathematical Society, 1999, p. 7.

Grigory Mikhalkin, Enumerative tropical algebraic geometry in R^2, arXiv:math/0312530v4. [From N. J. A. Sloane, Sep 27 2010]

Ian Strachan, How to count curves: from C. 19 problems to C. 20 solutions, Phil. Trans. Royal Soc. London, A 351 (2003), 2633-2647.

LINKS

T. D. Noe, Table of n, a(n) for n=1..50

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.

FORMULA

a_d = sum_{i+j=d} a_i*a_j ( i^2*j^2*binom(3d-4, 3i-2) - i^3*j*binom(3d-4, 3i-1) ).

MAPLE

a := proc(d:nonnegint) options remember; if d = 1 then 1 else sum('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

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}] (* From 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)! ))

CROSSREFS

Sequence in context: A159722 A042111 A159644 * A126159 A071307 A195574

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.

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Puzzles | Hot | Classics
Recent Additions | More pages | Superseeker | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified February 13 21:09 EST 2012. Contains 205561 sequences.