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

1,4

Links

Table of n, a(n) for n=1..18.

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).

Formula

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

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 *)

Crossrefs

Cf. A013587.

Sequence in context: A158114 A082736 A157039 * A203313 A087178 A104595

Adjacent sequences: A171114 A171115 A171116 * A171118 A171119 A171120

Keyword

nonn

Author

N. J. A. Sloane, Sep 27 2010

Extensions

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

Status

approved