A171113 -id:A171113 - cp's OEIS frontend

A171108 a(n) is the Severi degree for curves of degree n and cogenus 2.

0, 0, 21, 225, 882, 2370, 5175, 9891, 17220, 27972, 43065, 63525, 90486, 125190, 168987, 223335, 289800, 370056, 465885, 579177, 711930, 866250, 1044351, 1248555, 1481292, 1745100, 2042625, 2376621, 2749950, 3165582, 3626595, 4136175, 4697616, 5314320

Offset: 1

N. J. A. Sloane, Sep 27 2010

Severi degree N(n, delta) is the number of degree n plane curves which have delta nodes and pass through a generic configuration of n*(n+3)/2-delta points on the plane. delta is called the cogenus of these curves. See Fomin and Mikhalkin (2010), Section 1.2 "Combinatorial rules for Gromov-Witten invariants and Severi degrees" and 5 "Node polynomials". - Andrey Zabolotskiy, Jan 18 2021

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 (see first page).
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 (5,-10,10,-5,1).

Severi degrees N(n, delta) for other values of delta: A033428(n-1) (1), A171113 (3), A328551 (4), A328552 (5), A171116 (6).

Table[3(n-1)(n-2)(3n^2-3n-11)/2,{n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{0,0,21,225,882},40] (* Harvey P. Dale, Feb 01 2013 *)

concat([0,0], Vec(3*x^3*(7 + 40*x - 11*x^2) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Nov 01 2019

a(n) = 3*(n-1)*(n-2)*(3*n^2-3*n-11)/2.
a(1)=0, a(2)=0, a(3)=21, a(4)=225, a(5)=882, a(n) = 5*a(n-1)-10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Feb 01 2013
G.f.: 3*x^2*(-7-40*x+11*x^2) / (x-1)^5 . - R. J. Mathar, Dec 19 2013

New name from Andrey Zabolotskiy, Jan 18 2021

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

Original entry on oeis.org

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

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A171108 a(n) is the Severi degree for curves of degree n and cogenus 2.

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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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