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

%I #23 Jan 19 2021 05:51:03

%S 0,0,0,378,90027,2931831,33720354,224710119,1068797961,4037126346,

%T 12886585236,36161763120,91629683271,213681907449,465104644470,

%U 955060713621,1865654931141,3490074060228,6286011239592,10948910130774,18510503248611,30469179410667

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

%C 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

%H Colin Barker, <a href="/A328552/b328552.txt">Table of n, a(n) for n = 1..1000</a>

%H Florian Block, <a href="https://arxiv.org/abs/1006.0218">Computing node polynomials for plane curves</a>, arXiv:1006.0218 [math.AG], 2010-2011; Math. Res. Lett. 18, (2011), no. 4, 621-643.

%H Sergey Fomin and Grigory Mikhalkin, <a href="https://doi.org/10.4171/JEMS/238">Labeled floor diagrams for plane curves</a>, Journal of the European Mathematical Society 012.6 (2010): 1453-1496; arXiv:<a href="https://arxiv.org/abs/0906.3828">0906.3828</a> [math.AG], 2009-2010.

%H Israel Vainsencher, <a href="https://arxiv.org/abs/alg-geom/9312012">Enumeration of n-fold tangent hyperplanes to a surface</a>, arXiv preprint alg-geom/9312012, 1993-1994; J. Algebraic Geom., 4 (1995), 503-526. See Section 5.1.2.

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

%F 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.

%F 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

%F 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

%o (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

%o (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

%Y Cf. A171108, A171113, A328551.

%K nonn,easy

%O 1,4

%A _N. J. A. Sloane_, Oct 29 2019

%E New name and a(1)=a(2)=a(3)=0 from _Andrey Zabolotskiy_, Jan 19 2021