A171108 -id:A171108 - cp's OEIS frontend

A171112 Duplicate of A171108.

0, 0, 21, 225, 882, 2370

Offset: 1

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A328551 a(n) is the Severi degree for curves of degree n and cogenus 4.

0, 0, 0, 666, 36975, 437517, 2667375, 11225145, 37206936, 104285790, 257991042, 579308220, 1203756165, 2347234131, 4340067705, 7670818467, 13041558390, 21436446060, 34205577876, 53166223470, 80723690667, 120014201385, 175072295955, 251025419421

Offset: 1

N. J. A. Sloane, Oct 27 2019

Setting n=4 gives a(4) = 666, and Vainsencher remarks that "... 666 = 126 + 540 [is] the number of 4-nodal quartics through 10 general points. Indeed, a plane quartic with 4 nodes splits as a union of 2 conics, 126 of which pass through 10 points, or of a singular cubic and a line through 10 points."

All terms are divisible by 3, all but every third by 9. - M. F. Hasler, Oct 30 2019

Colin Barker, Table of n, a(n) for n = 1..1000
Florian Block, Computing node polynomials for plane curves, arXiv:1006.0218 [math.AG], 2010-2011; Math. Res. Lett. 18, (2011), no. 4, 621-643.
Florian Block, Susan Jane Colley, and Gary Kennedy, Computing Severi degrees with long-edge graphs, Bulletin of the Brazilian Mathematical Society, New Series 45.4 (2014): 625-647. Also arXiv:1303.5308 [math.AG], 2013.
Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, Journal of the European Mathematical Society 012.6 (2010): 1453-1496; arXiv:0906.3828 [math.AG], 2009-2010.
Israel Vainsencher, Enumeration of n-fold tangent hyperplanes to a surface, arXiv:alg-geom/9312012, 1993-1994; J. Algebraic Geom., 4 (1995), 503-526. See Section 5.1.1.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A171108, A171113, A328552.

concat([0, 0, 0], Vec(3*x^4*(222 + 10327*x + 42906*x^2 + 1626*x^3 - 17534*x^4 + 9879*x^5 - 2226*x^6 + 160*x^7) / (1 - x)^9 + O(x^30))) \\ Colin Barker, Oct 28 2019

{A328551(n, c=[222, 11881, 109530, 378831, 632340, 555660, 249480, 45360], p=3)=sum(k=1,min(#c,n-=3), c[k]*p*=(n-k+1)/k)} \\ M. F. Hasler, Oct 30 2019

a(n) = -8865 + (18057/4)*n + (37881/8)*n^2 - 2529*n^3 - 642*n^4 + (1809/4)*n^5 - 27*n^7 + (27/8)*n^8 for n > 2.
From Colin Barker, Oct 28 2019: (Start)
G.f.: 3*x^4*(222 + 10327*x + 42906*x^2 + 1626*x^3 - 17534*x^4 + 9879*x^5 - 2226*x^6 + 160*x^7) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>11.
(End)
a(2+3k) == 3 (mod 9), all other a(n) == 0 (mod 9). Periods mod 5, 7, 2 (of length 5, 7, 8): a(3..7 + 5k) == (0, 1, 0, 2, 0) (mod 5). a(3..9 + 7k) == (0, 1, 1, 3, 4, 1, 4) (mod 7). If 1 <= m <= 8, then a(m + 8k) is odd iff m > 4. - M. F. Hasler, Oct 30 2019

New name and a(1)=a(2)=0 from Andrey Zabolotskiy, Jan 19 2021

A328552 a(n) is the Severi degree for curves of degree n and cogenus 5.

Original entry on oeis.org

0, 0, 0, 378, 90027, 2931831, 33720354, 224710119, 1068797961, 4037126346, 12886585236, 36161763120, 91629683271, 213681907449, 465104644470, 955060713621, 1865654931141, 3490074060228, 6286011239592, 10948910130774, 18510503248611, 30469179410667

Offset: 1

N. J. A. Sloane, Oct 29 2019

All terms are divisible by 9: (a(n)) = 9*(42, 10003, 325759, 3746706, 24967791, ...). Satisfies a linear recurrence with characteristic polynomial (x-1)^11. - M. F. Hasler, Oct 30 2019

Colin Barker, Table of n, a(n) for n = 1..1000
Florian Block, Computing node polynomials for plane curves, arXiv:1006.0218 [math.AG], 2010-2011; Math. Res. Lett. 18, (2011), no. 4, 621-643.
Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, Journal of the European Mathematical Society 012.6 (2010): 1453-1496; arXiv:0906.3828 [math.AG], 2009-2010.
Israel Vainsencher, Enumeration of n-fold tangent hyperplanes to a surface, arXiv preprint alg-geom/9312012, 1993-1994; J. Algebraic Geom., 4 (1995), 503-526. See Section 5.1.2.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A171108, A171113, A328551.

{A328552(n,c=[9961, 305795, 2799396, 11895551, 28175817, 40446774, 36208620, 19852560, 6123600, 816480], p=9)=if(n<4,0,sum(k=1,min(#c,n-=4),c[k]*p*=(n-k+1)/k,378))} \\ M. F. Hasler, Oct 30 2019

concat([0, 0, 0], Vec(9*x^4*(42 + 9541*x + 218036*x^2 + 706592*x^3 + 34135*x^4 - 290191*x^5 + 181478*x^6 - 45302*x^7 + 677*x^8 + 1664*x^9 - 192*x^10) / (1 - x)^11 + O(x^40))) \\ Colin Barker, Oct 30 2019

a(n) = (81/40)*n^10 - (81/4)*n^9 - (27/8)*n^8 + (2349/4)*n^7 - (1044)*n^6 - (127071/20)*n^5 + (128859/8)*n^4 + (59097/2)*n^3 - (3528381/40)*n^2 - (946929/20)*n + 153513 for n > 3.
G.f.: 9*x^4*(42 + 9541*x + 218036*x^2 + 706592*x^3 + 34135*x^4 - 290191*x^5 + 181478*x^6 - 45302*x^7 + 677*x^8 + 1664*x^9 - 192*x^10)/(1 - x)^11. - M. F. Hasler, Oct 30 2019
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n>9. - Colin Barker, Oct 30 2019

New name and a(1)=a(2)=a(3)=0 from Andrey Zabolotskiy, Jan 19 2021

A171113 a(n) is the Severi degree for curves of degree n and cogenus 3.

Original entry on oeis.org

0, 0, 15, 675, 7915, 41310, 145383, 404185, 959115, 2029980, 3939295, 7139823, 12245355, 20064730, 31639095, 48282405, 71625163, 103661400, 146798895, 203912635, 278401515, 374248278, 496082695, 649247985, 839870475, 1074932500, 1362348543, 1711044615

Offset: 1

N. J. A. Sloane, Sep 27 2010

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Florian Block, Computing node polynomials for plane curves, arXiv:1006.0218 [math.AG], 2010-2011; Math. Res. Lett. 18, (2011), no. 4, 621-643.
Florian Block, Susan Jane Colley, and Gary Kennedy, Computing Severi degrees with long-edge graphs, Bulletin of the Brazilian Mathematical Society, New Series 45.4 (2014): 625-647. Also arXiv:1303.5308 [math.AG], 2013 (see first page).
P. Di Francesco and C. Itzykson, Quantum Intersection Rings, in: The Moduli Space of Curves, Birkhäuser Boston, 1995; on arXiv, arXiv:hep-th/9412175, 1994. See Proposition 2 (iii) and the following Remark (a).
Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, Journal of the European Mathematical Society 012.6 (2010): 1453-1496; arXiv:0906.3828 [math.AG], 2009-2010.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A171108, A328551, A328552.

LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,15,675,7915,41310,145383,404185,959115},30] (* Harvey P. Dale, Jun 15 2021 *)

[0, 0] + [(9*d**6 + 9*d**4 + 423*d**3 - 829*d)//2 - 27*d**5 - 229*d**2 + 525 for d in range(3, 30)] # Andrey Zabolotskiy, Jan 12 2021

a(n) = 9*n^6/2 - 27*n^5 + 9*n^4/2 + 423*n^3/2 - 229*n^2 - 829*n/2 + 525 for n > 2. - Andrey Zabolotskiy, Jan 19 2021

Terms a(7) and beyond from Andrey Zabolotskiy, Jan 12 2021

A171116 a(n) is the Severi degree for curves of degree n and cogenus 6.

Original entry on oeis.org

0, 0, 0, 105, 109781, 12597900, 302280963, 3356773532, 23599355991, 122416062018, 510681301550, 1807308075111, 5622246678741, 15761274284852, 40547443860105, 97044388890450, 218379097055159, 465931135430250, 948922558475388, 1854955331788517, 3496355565562725

Offset: 1

N. J. A. Sloane, Sep 27 2010

Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, Journal of the European Mathematical Society 012.6 (2010): 1453-1496; arXiv:0906.3828 [math.AG], 2009-2010.
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A171108.

a[n_ ? (#<4&)] = 0;
a[n_] := 81/80 n^12 - 243/20 n^11 - 81/20 n^10 + 8667/16 n^9 - 9297/8 n^8 - 47727/5 n^7 + 2458629/80 n^6 + 3243249/40 n^5 - 6577679/20 n^4 - 25387481/80 n^3 + 6352577/4 n^2 + 8290623/20 n - 2699706;
Table[a[n], {n, 20}] (* Andrey Zabolotskiy, May 02 2022 *)

Name edited, terms a(7) and beyond added by Andrey Zabolotskiy, May 02 2022

A171105 Multicomponent Gromov-Witten invariants for genus 0.

Original entry on oeis.org

1, 1, 12, 675, 109781, 40047888

Offset: 1

N. J. A. Sloane, Sep 27 2010

In this entry and in A171104, a multicomponent Gromov-Witten invariant is the number of (possibly reducible, hence "multicomponent") curves in CP^2 of degree n and genus g passing through given 3n-1+g points, so this is the Severi degree N(n, delta) where cogenus delta = (n-1)*(n-2)/2 - g, cf. A171108 and references therein. In particular, a(5) = A171116(5). - Andrey Zabolotskiy, May 04 2022

Florian Block, Computing node polynomials for plane curves, arXiv:1006.0218 [math.AG], 2010-2011; Math. Res. Lett. 18, (2011), no. 4, 621-643. See Appendix B.
Grigory Mikhalkin, Enumerative tropical algebraic geometry in R^2, arXiv:math/0312530 [math.AG], 2003-2004.

Cf. A171104, A171108, A171116.

Cf. Gromov-Witten invariants, counting irreducible curves only: A171109, A171110, A171111.

a(5)-a(6) added by Andrey Zabolotskiy, May 04 2022

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A171112 Duplicate of A171108.

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A328551 a(n) is the Severi degree for curves of degree n and cogenus 4.

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A328552 a(n) is the Severi degree for curves of degree n and cogenus 5.

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A171113 a(n) is the Severi degree for curves of degree n and cogenus 3.

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A171116 a(n) is the Severi degree for curves of degree n and cogenus 6.

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A171105 Multicomponent Gromov-Witten invariants for genus 0.

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