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).

5, 2875, 609250, 317206375, 242467530000, 229305888887625, 248249742118022000, 295091050570845659250, 375632160937476603550000, 503840510416985243645106250, 704288164978454686113488249750, 1017913203569692432490203659468875, 1512323901934139334751675234074638000

Offset

0,1

Comments

These integers are actually instanton numbers (or BPS states degeneracies). - Daniel Grunberg (grunberg(AT)mpim-bonn.mpg.de), Aug 18 2004

Equal to the number of degree-n rational curves on a general quintic for n <= 9, but not for n = 10 (see A076912).

References

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. A. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Amer. Math. Soc., 1999.

Links

Gheorghe Coserea, Table of n, a(n) for n = 0..301 (first 101 terms from T. D. Noe)

V. Batyrev, Review of "Mirror Symmetry and Algebraic Geometry", by D. A. Cox and S. Katz, Bull. Amer. Math. Soc., 37 (No. 4, 2000), 473-476.

P. Candelas et al., A pair of Calabi-yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), 21-74.

R. H. Dijkgraaf, The Mathematics of String Theory, 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.

Steven R. Finch, Enumerative geometry, February 24, 2014. [Cached copy, with permission of the author]

Trygve Johnsen and Steven L. Kleiman, Rational curves of degree at most 9 on a general quintic threefold, arXiv:alg-geom/9510015, 1995.

Trygve Johnsen and Steven L. Kleiman, Toward Clemens' Conjecture in degrees between 10 and 24, arXiv:alg-geom/9601024, 1996.

B. Mazur, Perturbations, deformations and variations ..., Bull. Amer. Math. Soc., 41 (2004), 307-336.

David R. Morrison, Mathematical Aspects of Mirror Symmetry, arXiv:alg-geom/9609021, 1996; in Complex Algebraic Geometry (J. Kollár, ed.), IAS/Park City Math. Series, vol. 3, 1997, pp. 265-340.

R. Pandharipande, Rational curves on hypersurfaces (after A. Givental), Séminaire Bourbaki, Vol. 1997/98. Astérisque No. 252 (1998), Exp. No. 848, 5, 307-340.

Example

G.f. = 5 + 2875*x + 609250*x^2 + 317206375*x^3 + 242467530000*x^4 + ...

Mathematica

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 *)

Prog

(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 */
(PARI)
cumsum(v) = for(i=2, #v, v[i] += v[i-1]); v;
A060345_list(N) = {
  my(x = 'x + O('x^(N+1)), h = cumsum(vector(5*N, n, 1/n)),
     y0 = sum(n=0, N, (5*n)!/n!^5 * x^n),
     y1 = 5 * sum(n = 1, N, ((5*n)!/n!^5 * (h[5*n] - h[n])) * x^n),
     Qx = x * exp(y1/y0), Xq = serreverse(Qx));
  Vec(5 * (x * Xq'/Xq)^3 / ((1 - 3125*Xq) * sqr(subst(y0, 'x, Xq))));
};
seq(N) = {
  my(v1 = A060345_list(N+1),
     v2 = dirmul(vector(N, n, moebius(n)), vector(N, n, v1[n+1])));
  concat(5, vector(#v2, n, v2[n]/n^3));
};
seq(20)  \\ Gheorghe Coserea, Jul 28 2016

Crossrefs

Cf. A060345, A076912.

Sequence in context: A258980 A096934 A076912 * A199878 A076908 A060345

Adjacent sequences: A060038 A060039 A060040 * A060042 A060043 A060044

Keyword

nonn,nice

Author

N. J. A. Sloane, Mar 19 2001

Status

approved