A060345
An expansion related to Yukawa coupling.
Original entry on oeis.org
5, 2875, 4876875, 8564575000, 15517926796875, 28663236110956000, 53621944306062201000, 101216230345800061125625, 192323666400003538944396875, 367299732093982242625847031250, 704288164978454714776724365580000, 1354842473951260627644461070753075500, 2613295702542192770504516764304958585000
Offset: 0
a(10) = A060041(1) + 8*A060041(2) + 125*A060041(5) + 1000*A060041(10) = 704288164978454714776724365580000.
- Gheorghe Coserea, Table of n, a(n) for n = 0..300
- P. Candelas et al., A pair of Calabi-yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), 21-74.
- Daniel B. Grunberg and Pieter Moree, with an Appendix by Don Zagier, Sequences of enumerative geometry: congruences and asymptotics, arXiv:math/0610286 [math.NT], 2006.
- David R. Morrison, Mathematical Aspects of Mirror Symmetry, arXiv:alg-geom/9609021, 1996, see Table 1 p. 60; in Complex Algebraic Geometry (J. Kollár, ed.), IAS/Park City Math. Series, vol. 3, 1997, pp. 265-340.
-
cumsum(v) = for(i=2, #v, v[i] += v[i-1]); v;
seq(N, {d=5}) = {
my(x = 'x + O('x^(N+1)), h = cumsum(vector(d*N, n, 1/n)),
y0 = sum(n=0, N, (d*n)!/n!^d * x^n),
y1 = d * sum(n = 1, N, ((d*n)!/n!^d * (h[d*n] - h[n])) * x^n),
Qx = x * exp(y1/y0), Xq = serreverse(Qx));
Vec(d * (x * Xq'/Xq)^(d-2) / ((1 - d^d*Xq) * sqr(subst(y0, 'x, Xq))));
};
seq(20) \\ Gheorghe Coserea, Jul 29 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
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.
a(10) =
A060041(10) - 6 * 17601000 added by
Andrey Zabolotskiy, Sep 10 2022 (see Encyclopedia of Mathematics, Clemens' conjecture)
A090005
Gromov-Witten invariants of intersection type (3,3).
Original entry on oeis.org
1053, 52812, 6424326, 1139448384, 249787892583, 62660964509532, 17256453900822009, 5088842568426162960, 1581250717976557887945, 512045241907209106828608
Offset: 1
- 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.
A090008
Gromov-Witten invariants of intersection type (2,2,3).
Original entry on oeis.org
720, 22428, 1611504, 168199200, 21676931712, 3195557904564, 517064870788848, 89580965599606752, 16352303769375910848, 3110686153486233022944
Offset: 1
- 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.
A090006
Gromov-Witten invariants of intersection type (2,4).
Original entry on oeis.org
1280, 92288, 15655168, 3883902528, 1190923282176, 417874605342336, 160964588281789696, 66392895625625639488, 28855060316616488359936, 13069047760169269024822656
Offset: 1
- 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.
A090007
Gromov-Witten invariants of intersection type (2,2,2,2).
Original entry on oeis.org
512, 9728, 416256, 25703936, 1957983744, 1705359223200, 16300354777600, 1668063096387072, 179845756064329728, 20206497983891554816
Offset: 1
- 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.
A308279
The weighted sum of BPS states n_1^n for the quintic.
Original entry on oeis.org
0, 0, 609250, 3721431625, 12129909700200, 31147299733286500, 71578406022880761750, 154990541752961568418125, 324064464310279585657008750, 662863774391414096742406576300, 1336442091735463067608016312923750, 2668147157687867719032781403203942875, 5290387928421738826068606777043538125500
Offset: 1
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