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
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane, Mar 19 2001
STATUS
approved