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

1,4

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).

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

Prog

(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

Crossrefs

Cf. A171108, A171113, A328552.

Sequence in context: A057564 A227490 A171114 * A172922 A046694 A326255

Adjacent sequences: A328548 A328549 A328550 * A328552 A328553 A328554

Keyword

nonn,easy

Author

N. J. A. Sloane, Oct 27 2019

Extensions

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

Status

approved