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

Author

N. J. A. Sloane, Oct 29 2019

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A171108, A171113, A328551.

Programs

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

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

Extensions

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