A171117 A particular case of Gromov-Witten numbers: a(n) is the number of complex rational curves of degree n and genus 0 in CP^3 passing through 2n given points.

1, 0, 1, 4, 105, 2576, 122129, 7397760, 629336977, 68265049600, 9386419113537, 1583207240397824, 322519291535862713, 77985053716765181952, 22094670475785827572945, 7249172440569540585914368, 2727206213196927179246863137, 1166222035906526210266584842240

Offset: 1

Author

N. J. A. Sloane, Sep 27 2010

Keywords

• nonn

Links

• 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.
• Penka Georgieva and Aleksey Zinger, Enumeration of real curves in CP^{2n-1} and a Witten-Dijkgraaf-Verlinde-Verlinde relation for real Gromov-Witten invariants, Duke Math. J., 166 (2017), 3291-3347; arXiv:1309.4079 [math.SG], 2013-2017. See Eq. (7.1).

Crossrefs

Cf. A013587.

Programs

Mathematica

n[1] = nt[1] = 1;
n[d_] := n[d] = Sum[With[{d2 = d - d1}, (d2^2 Binomial[2 d - 3, 2 d1 - 2] - d1 d2 Binomial[2 d - 3, 2 d1 - 1]) nt[d1] n[d2]], {d1, d - 1}];
nt[d_] := nt[d] = d n[d] + Sum[With[{d2 = d - d1}, (d1 d2^2 Binomial[2 d - 2, 2 d1 - 1] - d2^3 Binomial[2 d - 2, 2 d1 - 2]) nt[d1] n[d2]], {d1, d - 1}];
Table[n[d], {d, 20}] (* Andrey Zabolotskiy, May 03 2022 *)

Formula

a(n) ~ c * d^n * n^(2*n-3), where d = 0.22437689379499207235291475487670864472074175469311760751181993..., c = 2.114876309952735589169436238081913983666848627651832555153... - Vaclav Kotesovec, Apr 28 2024

Extensions

Name edited, terms a(8) and beyond added by Andrey Zabolotskiy, May 03 2022