A076912 -id:A076912 - 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).

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

Offset: 0

N. J. A. Sloane, Mar 19 2001

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).

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

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.

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.

Cf. A060345, A076912.

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 *)

{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 */

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

A027363 Generalizing the 27 lines on a cubic surface: number of lines on the generic hypersurface of degree 2n-1 in complex projective (n+1)-space.

Original entry on oeis.org

1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775, 289139638632755625, 520764738758073845321, 1192221463356102320754899, 3381929766320534635615064019, 11643962664020516264785825991165

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it), Oct 15 1997

Van der Waerden, see one of his 'Zur algebraischen Geometrie' papers.

Gheorghe Coserea, Table of n, a(n) for n = 1..300
Steven R. Finch, Enumerative geometry, February 24, 2014. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 752.
Daniel B. Grunberg and Pieter Moree, with an Appendix by Don Zagier, Sequences of enumerative geometry: congruences and asymptotics, arXiv math.NT/0610286, 2006.

Cf. A013587, A076912.

a[n_] := Coefficient[ (1-x)*Product[ 2n-1-j+j*x, {j, 0, 2n-1}], x, n]; Table[a[n], {n, 1, 12}] (* Jean-François Alcover, Jan 23 2012, from second formula *)

a(n) = my(x='x); polcoeff((1-x) * prod(j=0, 2*n-1, 2*n-1-j + j*x), n);
vector(20, n, a(n))  \\ Gheorghe Coserea, Jul 28 2016

Let b(n, i)=i/(n-i+1) and g(n, k)=s[ k ](b(n, 1), b(n, 2), ..., b(n, n)), where s[ k ] is the k-th elementary symmetric function; a(n) = (2n-1)^2 * (2n-2)! * [ g(2n-2, n-1) - g(2n-2, n) ].
a(n) = [x^n] (1-x)*Product_{j=0..2n-1}(2n-1-j+j*x). [Van der Waerden]
a(n) ~ sqrt(27/Pi) * (2*n-1)^(2*n-3/2) * (1-9/(8*n)+O(1/n^2)). - Gheorghe Coserea, Jul 28 2016

A076913 Coefficients of 3-point function in dimension 4.

Original entry on oeis.org

6, 60480, 440884080, 6255156277440, 117715791990353760, 2591176156368821985600

Offset: 0

N. J. A. Sloane, Nov 28 2002

nonn

Klemm and Pandharipande's Table 2 contains the sequence that agrees with the initial terms given here, a(1)-a(5). It continues 63022367592536650014764880, 1642558496795158117310144372160, 45038918271966862868230872208340160. - Andrey Zabolotskiy, Sep 11 2022

Geir Ellingsrud and Stein Arild Stromme, Bott's formula and enumerative geometry. J. Amer. Math. Soc. 9 (1996), 175-193. [arXiv:alg-geom/9411005]
A. Klemm and R. Pandharipande, Enumerative geometry of Calabi-Yau 4-folds, Commun. Math. Phys., 281 (2008), 621-653; arXiv:math/0702189 [math.AG], 2007.
David R. Morrison, Mathematical Aspects of Mirror Symmetry, in Complex Algebraic Geometry (J. Kollár, ed.), IAS/Park City Math. Series, vol. 3, 1997, pp. 265-340.

Cf. A060345, A076909, A076910, A076911, A076912, A076914, A076915, A076916, A076917, A076923.

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A096934 Duplicate of A076912.

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A199878 Erroneous version of A076912.

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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).

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A027363 Generalizing the 27 lines on a cubic surface: number of lines on the generic hypersurface of degree 2n-1 in complex projective (n+1)-space.

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A076913 Coefficients of 3-point function in dimension 4.

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