A013587 -id:A013587 - 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

A076912 Number of degree-n rational curves on a general quintic threefold.

Original entry on oeis.org

5, 2875, 609250, 317206375, 242467530000, 229305888887625, 248249742118022000, 295091050570845659250, 375632160937476603550000, 503840510416985243645106250, 704288164978454686113382643750

Offset: 0

N. J. A. Sloane, Nov 28 2002

nonn

			a(1) = 2875 = number of lines in the quintic.

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.
Ellingsrud, Geir and Stromme, Stein Arild, The number of twisted cubic curves on the general quintic threefold (preliminary version). In Essays on Mirror Manifolds, 181-222, Int. Press, Hong Kong, 1992.

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.
Geir Ellingsrud and Stein Arild Stromme, The number of twisted cubic curves on the general quintic threefold, Math. Scand. 76 (1995), no. 1, 5-34.
Geir Ellingsrud and Stein Arild Stromme, Bott's formula and enumerative geometry, J. Amer. Math. Soc. 9 (1996), 175-193; arXiv:alg-geom/9411005, 1994.
Encyclopedia of Mathematics, Clemens' conjecture.
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, 1995-1996.
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 (and "near-misses") in geometry, physics, and number theory, 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.
Wikipedia, Quintic threefold
Shing-Tung Yau and Steve Nadis, String Theory and the Geometry of the Universe's Hidden Dimensions, Notices Amer. Math. Soc., 58 (Sep 2011), 1067-1076.

Coincides with A060041 for n <= 9, but not for n = 10.

Cf. A060345, A076913.

a(10) = A060041(10) - 6 * 17601000 added by Andrey Zabolotskiy, Sep 10 2022 (see Encyclopedia of Mathematics, Clemens' conjecture)

A171109 Gromov-Witten invariants for genus 1.

Original entry on oeis.org

0, 0, 1, 225, 87192, 57435240, 60478511040, 96212546526096, 220716443548094400, 702901008498298112640, 3011788599493603375929600, 16916605752011965307094124800, 121848941490162387021464335349760, 1104617766019213143798099163667712000

Offset: 1

N. J. A. Sloane, Sep 27 2010

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..150
Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, arXiv:0906.3828 [math.AG], 2009-2010.
Rahul Pandharipande, A geometric construction of Getzler's relation, arXiv:alg-geom/9705016, 1997.

Cf. A013587, A171110, A171111.

(* b = A013587 *) b[n_] := b[n] = If[n==1, 1, Sum[b[k] b[n-k] k^2 (n-k) (3k-n) (3n-4)!/(3k-1)!/(3(n-k)-2)!, {k, 1, n-1}]];
a[n_] := a[n] = Module[{t1, t2}, t1 = Binomial[n, 3] b[n]; t2 = Sum[ Binomial[3n-1, 3k-1](3k^2-2k)(n-k) b[k] a[n-k], {k, n-1}]; t1/12 + t2/9];
Array[a, 14] (* Jean-François Alcover, Oct 08 2018, after Gheorghe Coserea *)

A013587_seq(N) = {
  my(a = vector(N), t1, t2); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1,
    t1 = binomial(3*n-4, 3*k-2)*(k*(n-k))^2;
    t2 = binomial(3*n-4,3*k-1)*k^3*(n-k);
    (t1 - t2)*a[k]*a[n-k])); a;
};
A171109_seq(N) = {
  my(a = vector(N), b=A013587_seq(N), t1, t2);
  for (n=3, N, t1 = binomial(n,3)*b[n];
    t2 = sum(k=1,n-1,binomial(3*n-1,3*k-1)*(3*k^2-2*k)*(n-k)*b[k]*a[n-k]);
    a[n] = (t1/12 + t2/9)); a;
};
A171109_seq(14) \\ Gheorghe Coserea, Jan 01 2018

Terms a(7) and beyond from Gheorghe Coserea, Jan 01 2018

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

A319851 Welschinger invariant for the number of real plane curves of degree n passing through 3*n-1 general points.

Original entry on oeis.org

1, 1, 8, 240, 18264, 2845440, 792731520, 359935488000, 248962406889600

Offset: 1

Georges Perrotte, Sep 29 2018

a(n) is the Welschinger invariant #n.

Aubin Arroyo, Erwan Brugallé and Lucía López de Medrano, Recursive formulas for Welschinger invariants of the projective plane, International Mathematics Research Notices, 2011, 1107-1134; arXiv:0809.1541 [math.AG], 2008-2010. See numbers W2(n,0) in Section 7.3.
Erwan Brugallé, Géométries énumératives complexe, réelle et tropicale, Journées mathématiques X-UPS, École polytechnique, 2008. See Table 3, p. 54.
Antoine Chambert-Loir, Quand la géométrie devient tropicale, Pour la Science, No 492, October 2018 (in French).
I. Itenberg, V. Kharlamov & E. Shustin, Welschinger invariant and enumeration of real rational curves, Int. Math. Res. Not. (2003), no. 49, pp. 2639-2653.
I. Itenberg, V. Kharlamov & E. Shustin, Welschinger invariant and enumeration of real rational curves, arXiv:math/0303378 [math.AG], 2003.
I. Itenberg, V. Kharlamov & E. Shustin, Logarithmic equivalence of the Welschinger and the Gromov-Witten invariants, Uspekhi Mat. Nauk 59 (2004), no. 6(360), pp. 85-110.
I. Itenberg, V. Kharlamov & E. Shustin, Logarithmic equivalence of the Welschinger and the Gromov-Witten invariants, arXiv:math/0407188 [math.AG], 2004.

Cf. A013587, A171118.

a(8)-a(9) added from Arroyo et al. and name clarified by Andrey Zabolotskiy, May 03 2022, based on contribution by Michel Marcus

A238370 Number of rational curves on S of degree d passing through d-1 general points, where S is a general cubic surface in projective 3-space.

Original entry on oeis.org

27, 27, 72, 216, 459, 936

Offset: 1

Steven Finch, Feb 25 2014

Finding a recursive formula (à la Kontsevich for A013587?) remains open.

The famous 27 lines on a cubic surface is the case a(1) = 27. - Michael Somos, Feb 25 2014

Steven R. Finch, Enumerative geometry, February 24, 2014. [Cached copy, with permission of the author]
W. Traves, Conics on the cubic surface, MAAC 2004, George Mason University.

Cf. A013587, A027363.

A171117 A particular case of Gromov-Witten numbers: a(n) is the number of complex rational curves of degree n and genus 0 in CP^3 passing through 2n given points.

Original entry on oeis.org

1, 0, 1, 4, 105, 2576, 122129, 7397760, 629336977, 68265049600, 9386419113537, 1583207240397824, 322519291535862713, 77985053716765181952, 22094670475785827572945, 7249172440569540585914368, 2727206213196927179246863137, 1166222035906526210266584842240

Offset: 1

N. J. A. Sloane, Sep 27 2010

nonn

Erwan Brugallé and Grigory Mikhalkin, Enumeration of curves via floor diagrams, C. R. Acad. Sci. Paris, Ser. I, 345 (2007), 329-334; arXiv:0706.0083 [math.AG], 2007.
Penka Georgieva and Aleksey Zinger, Enumeration of real curves in CP^{2n-1} and a Witten-Dijkgraaf-Verlinde-Verlinde relation for real Gromov-Witten invariants, Duke Math. J., 166 (2017), 3291-3347; arXiv:1309.4079 [math.SG], 2013-2017. See Eq. (7.1).

Cf. A013587.

n[1] = nt[1] = 1;
n[d_] := n[d] = Sum[With[{d2 = d - d1}, (d2^2 Binomial[2 d - 3, 2 d1 - 2] - d1 d2 Binomial[2 d - 3, 2 d1 - 1]) nt[d1] n[d2]], {d1, d - 1}];
nt[d_] := nt[d] = d n[d] + Sum[With[{d2 = d - d1}, (d1 d2^2 Binomial[2 d - 2, 2 d1 - 1] - d2^3 Binomial[2 d - 2, 2 d1 - 2]) nt[d1] n[d2]], {d1, d - 1}];
Table[n[d], {d, 20}] (* Andrey Zabolotskiy, May 03 2022 *)

a(n) ~ c * d^n * n^(2*n-3), where d = 0.22437689379499207235291475487670864472074175469311760751181993..., c = 2.114876309952735589169436238081913983666848627651832555153... - Vaclav Kotesovec, Apr 28 2024

Name edited, terms a(8) and beyond added by Andrey Zabolotskiy, May 03 2022

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).

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A076912 Number of degree-n rational curves on a general quintic threefold.

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A171109 Gromov-Witten invariants for genus 1.

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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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A319851 Welschinger invariant for the number of real plane curves of degree n passing through 3*n-1 general points.

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A238370 Number of rational curves on S of degree d passing through d-1 general points, where S is a general cubic surface in projective 3-space.

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A171117 A particular case of Gromov-Witten numbers: a(n) is the number of complex rational curves of degree n and genus 0 in CP^3 passing through 2n given points.

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