0,1

These integers are actually instanton numbers (or PBS 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 (if Clemens's conjecture is true) not for n = 10 (see A076912).

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.

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

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

R. H. Dijkgraaf, The Mathematics of String Theory, pp. 58ff in "Aspects De La Physique En 2005: Einstein 1905-2005", Numero special de la Gazette des mathematiciens. Supplement au no. 106, Oct 2005, Societe Mathematique de France, Paris.

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

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

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

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

T. D. Noe, Table of n, a(n) for n=0..100

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.

David R. Morrison, Mathematical Aspects of Mirror Symmetry, in Complex Algebraic Geometry (J. Koll\'ar, ed.), IAS/Park City Math. Series, vol. 3, 1997, pp. 265-340

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\[Equal]s2]; t=Table[n[d]/.sol, {d, 1, nn}]//Flatten; - Daniel Grunberg (grunberg(AT)mpim-bonn.mpg.de), Aug 18 2004.

(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.

Cf. A060345, A079912.

Adjacent sequences: A060038 A060039 A060040 this_sequence A060042 A060043 A060044

Sequence in context: A117011 A096934 A076912 this_sequence A060345 A076908 A013782

nonn,nice

njas, Mar 19 2001