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

Author

N. J. A. Sloane, Oct 27 2019

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

Cf. A171108, A171113, A328552.

Programs

PARI

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

PARI

{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

Formula

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

Extensions

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