A171109 Gromov-Witten invariants for genus 1.

0, 0, 1, 225, 87192, 57435240, 60478511040, 96212546526096, 220716443548094400, 702901008498298112640, 3011788599493603375929600, 16916605752011965307094124800, 121848941490162387021464335349760, 1104617766019213143798099163667712000

Offset: 1

Author

N. J. A. Sloane, Sep 27 2010

Keywords

• nonn

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..150
• Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, arXiv:0906.3828 [math.AG], 2009-2010.
• Rahul Pandharipande, A geometric construction of Getzler's relation, arXiv:alg-geom/9705016, 1997.

Crossrefs

Cf. A013587, A171110, A171111.

Programs

Mathematica

(* b = A013587 *) b[n_] := b[n] = If[n==1, 1, Sum[b[k] b[n-k] k^2 (n-k) (3k-n) (3n-4)!/(3k-1)!/(3(n-k)-2)!, {k, 1, n-1}]];
a[n_] := a[n] = Module[{t1, t2}, t1 = Binomial[n, 3] b[n]; t2 = Sum[ Binomial[3n-1, 3k-1](3k^2-2k)(n-k) b[k] a[n-k], {k, n-1}]; t1/12 + t2/9];
Array[a, 14] (* Jean-François Alcover, Oct 08 2018, after Gheorghe Coserea *)

PARI

A013587_seq(N) = {
  my(a = vector(N), t1, t2); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1,
    t1 = binomial(3*n-4, 3*k-2)*(k*(n-k))^2;
    t2 = binomial(3*n-4,3*k-1)*k^3*(n-k);
    (t1 - t2)*a[k]*a[n-k])); a;
};
A171109_seq(N) = {
  my(a = vector(N), b=A013587_seq(N), t1, t2);
  for (n=3, N, t1 = binomial(n,3)*b[n];
    t2 = sum(k=1,n-1,binomial(3*n-1,3*k-1)*(3*k^2-2*k)*(n-k)*b[k]*a[n-k]);
    a[n] = (t1/12 + t2/9)); a;
};
A171109_seq(14) \\ Gheorghe Coserea, Jan 01 2018

Extensions

Terms a(7) and beyond from Gheorghe Coserea, Jan 01 2018