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

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

Offset

1,4

Comments

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

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.

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).

Formula

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

Prog

(PARI) {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
(PARI) 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

Crossrefs

Cf. A171108, A171113, A328551.

Sequence in context: A221803 A289347 A171115 * A225994 A228554 A108385

Adjacent sequences: A328549 A328550 A328551 * A328553 A328554 A328555

Keyword

nonn,easy

Author

N. J. A. Sloane, Oct 29 2019

Extensions

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

Status

approved