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A171109 Gromov-Witten invariants for genus 1.

%I A171109 #30 May 02 2022 17:27:57
%S A171109 0,0,1,225,87192,57435240,60478511040,96212546526096,
%T A171109 220716443548094400,702901008498298112640,3011788599493603375929600,
%U A171109 16916605752011965307094124800,121848941490162387021464335349760,1104617766019213143798099163667712000
%N A171109 Gromov-Witten invariants for genus 1.
%H A171109 Gheorghe Coserea, <a href="/A171109/b171109.txt">Table of n, a(n) for n = 1..150</a>
%H A171109 Sergey Fomin and Grigory Mikhalkin, <a href="http://arxiv.org/abs/0906.3828">Labeled floor diagrams for plane curves</a>, arXiv:0906.3828 [math.AG], 2009-2010.
%H A171109 Rahul Pandharipande, <a href="http://arxiv.org/abs/alg-geom/9705016">A geometric construction of Getzler's relation</a>, arXiv:alg-geom/9705016, 1997.
%t A171109 (* 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}]];
%t A171109 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];
%t A171109 Array[a, 14] (* _Jean-François Alcover_, Oct 08 2018, after _Gheorghe Coserea_ *)
%o A171109 (PARI)
%o A171109 A013587_seq(N) = {
%o A171109   my(a = vector(N), t1, t2); a[1] = 1;
%o A171109   for (n=2, N, a[n] = sum(k=1, n-1,
%o A171109     t1 = binomial(3*n-4, 3*k-2)*(k*(n-k))^2;
%o A171109     t2 = binomial(3*n-4,3*k-1)*k^3*(n-k);
%o A171109     (t1 - t2)*a[k]*a[n-k])); a;
%o A171109 };
%o A171109 A171109_seq(N) = {
%o A171109   my(a = vector(N), b=A013587_seq(N), t1, t2);
%o A171109   for (n=3, N, t1 = binomial(n,3)*b[n];
%o A171109     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]);
%o A171109     a[n] = (t1/12 + t2/9)); a;
%o A171109 };
%o A171109 A171109_seq(14) \\ _Gheorghe Coserea_, Jan 01 2018
%Y A171109 Cf. A013587, A171110, A171111.
%K A171109 nonn
%O A171109 1,4
%A A171109 _N. J. A. Sloane_, Sep 27 2010
%E A171109 Terms a(7) and beyond from _Gheorghe Coserea_, Jan 01 2018