cp's OEIS Frontend

A060041 Certain numbers a(n) related to Gromov-Witten invariants N_n in dimension n (see formula (7.45) on p. 202 of Cox and Katz).

%I A060041 #61 Nov 02 2023 02:03:30
%S A060041 5,2875,609250,317206375,242467530000,229305888887625,
%T A060041 248249742118022000,295091050570845659250,375632160937476603550000,
%U A060041 503840510416985243645106250,704288164978454686113488249750,1017913203569692432490203659468875,1512323901934139334751675234074638000
%N A060041 Certain numbers a(n) related to Gromov-Witten invariants N_n in dimension n (see formula (7.45) on p. 202 of Cox and Katz).
%C A060041 These integers are actually instanton numbers (or BPS states degeneracies). - Daniel Grunberg (grunberg(AT)mpim-bonn.mpg.de), Aug 18 2004
%C A060041 Equal to the number of degree-n rational curves on a general quintic for n <= 9, but not for n = 10 (see A076912).
%D A060041 J. Bertin and C. Peters, Variations of Hodge structure ..., pp. 151-232 of J. Bertin et al., eds., Introduction to Hodge Theory, Amer. Math. Soc. and Soc. Math. France, 2002; see p. 220.
%D A060041 D. A. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Amer. Math. Soc., 1999.
%H A060041 Gheorghe Coserea, <a href="/A060041/b060041.txt">Table of n, a(n) for n = 0..301</a> (first 101 terms from T. D. Noe)
%H A060041 V. Batyrev, <a href="http://www.ams.org/journal-getitem?pii=S0273-0979-00-00875-2">Review of "Mirror Symmetry and Algebraic Geometry"</a>, by D. A. Cox and S. Katz, Bull. Amer. Math. Soc., 37 (No. 4, 2000), 473-476.
%H A060041 P. Candelas et al., <a href="http://dx.doi.org/10.1016/0550-3213(91)90292-6">A pair of Calabi-yau manifolds as an exactly soluble superconformal theory</a>, Nuclear Phys. B 359 (1991), 21-74.
%H A060041 R. H. Dijkgraaf, <a href="http://www.bourbaphy.fr/dijkgraaf.pdf">The Mathematics of String Theory</a>, pp. 58ff in "Aspects De La Physique En 2005: Einstein 1905-2005", Numéro spécial de la Gazette des mathématiciens. Supplément au no. 106, Oct 2005, Société Mathématique de France, Paris.
%H A060041 Steven R. Finch, <a href="/A013587/a013587.pdf">Enumerative geometry</a>, February 24, 2014. [Cached copy, with permission of the author]
%H A060041 Trygve Johnsen and Steven L. Kleiman, <a href="http://arxiv.org/abs/alg-geom/9510015">Rational curves of degree at most 9 on a general quintic threefold</a>, arXiv:alg-geom/9510015, 1995.
%H A060041 Trygve Johnsen and Steven L. Kleiman, <a href="http://arxiv.org/abs/alg-geom/9601024">Toward Clemens' Conjecture in degrees between 10 and 24</a>, arXiv:alg-geom/9601024, 1996.
%H A060041 B. Mazur, <a href="http://dx.doi.org/10.1090/S0273-0979-04-01024-9">Perturbations, deformations and variations ...</a>, Bull. Amer. Math. Soc., 41 (2004), 307-336.
%H A060041 David R. Morrison, <a href="http://arXiv.org/abs/alg-geom/9609021">Mathematical Aspects of Mirror Symmetry</a>, arXiv:alg-geom/9609021, 1996; in Complex Algebraic Geometry (J. Kollár, ed.), IAS/Park City Math. Series, vol. 3, 1997, pp. 265-340.
%H A060041 R. Pandharipande, <a href="http://www.numdam.org/book-part/SB_1997-1998__40__307_0/">Rational curves on hypersurfaces (after A. Givental)</a>, Séminaire Bourbaki, Vol. 1997/98. Astérisque No. 252 (1998), Exp. No. 848, 5, 307-340.
%e A060041 G.f. = 5 + 2875*x + 609250*x^2 + 317206375*x^3 + 242467530000*x^4 + ...
%t A060041 nn=20; y0[x_]:=Sum[(5n)!/(n!)^5 x^n, {n, 0, nn}]; y1[x_]:=Sum[((5n)!/(n!)^5 5 Sum[1/j, {j, n+1, 5n}]) x^n, {n, 0, nn}]; qq=Series[x Exp[y1[x]/y0[x]], {x, 0, nn}]; x[q_]=InverseSeries[qq, q]; s1=(q/x[q] D[x[q], q])^3 5/((1-5^5 x[q]) y0[x[q]]^2); s2=Series[5+Sum[n[d] d^3 q^d/(1-q^d), {d, 1, nn}], {q, 0, nn}]; sol=Solve[s1==s2]; t=Table[n[d]/.sol, {d, 1, nn}]//Flatten; (* Daniel Grunberg (grunberg(AT)mpim-bonn.mpg.de), Aug 18 2004 *)
%o A060041 (PARI) {a(n) = local(A1, A2, A3); if( n<1, 5*(n==0), A1 = sum( k=0, n, (5*k)! / k!^5 * (-x)^k, x * O(x^n)); A2 = -x * exp(5 / A1 * sum( k=0, n, (sum( i=1, 5*k, 1/i) - sum( i=1, k, 1/i)) * (5*k)! / k!^5 * (-x)^k, x * O(x^n))); A3 = subst(5 / A1^2 / (1 + 5^5*x) / (x * A2'/A2)^3, x, serreverse(A2)); sumdiv( n, k, moebius(n / k) * polcoeff(A3, k))/n^3)}; /* _Michael Somos_, Mar 27 2004 */
%o A060041 (PARI)
%o A060041 cumsum(v) = for(i=2, #v, v[i] += v[i-1]); v;
%o A060041 A060345_list(N) = {
%o A060041   my(x = 'x + O('x^(N+1)), h = cumsum(vector(5*N, n, 1/n)),
%o A060041      y0 = sum(n=0, N, (5*n)!/n!^5 * x^n),
%o A060041      y1 = 5 * sum(n = 1, N, ((5*n)!/n!^5 * (h[5*n] - h[n])) * x^n),
%o A060041      Qx = x * exp(y1/y0), Xq = serreverse(Qx));
%o A060041   Vec(5 * (x * Xq'/Xq)^3 / ((1 - 3125*Xq) * sqr(subst(y0, 'x, Xq))));
%o A060041 };
%o A060041 seq(N) = {
%o A060041   my(v1 = A060345_list(N+1),
%o A060041      v2 = dirmul(vector(N, n, moebius(n)), vector(N, n, v1[n+1])));
%o A060041   concat(5, vector(#v2, n, v2[n]/n^3));
%o A060041 };
%o A060041 seq(20)  \\ _Gheorghe Coserea_, Jul 28 2016
%Y A060041 Cf. A060345, A076912.
%K A060041 nonn,nice
%O A060041 0,1
%A A060041 _N. J. A. Sloane_, Mar 19 2001