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%I #31 May 02 2022 17:27:52
%S 0,0,21,225,882,2370,5175,9891,17220,27972,43065,63525,90486,125190,
%T 168987,223335,289800,370056,465885,579177,711930,866250,1044351,
%U 1248555,1481292,1745100,2042625,2376621,2749950,3165582,3626595,4136175,4697616,5314320
%N a(n) is the Severi degree for curves of degree n and cogenus 2.
%C Severi degree N(n, delta) is the number of degree n plane curves which have delta nodes and pass through a generic configuration of n*(n+3)/2-delta points on the plane. delta is called the cogenus of these curves. See Fomin and Mikhalkin (2010), Section 1.2 "Combinatorial rules for Gromov-Witten invariants and Severi degrees" and 5 "Node polynomials". - _Andrey Zabolotskiy_, Jan 18 2021
%H Colin Barker, <a href="/A171108/b171108.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 Florian Block, Susan Jane Colley, and Gary Kennedy, <a href="https://arxiv.org/abs/1303.5308">Computing Severi degrees with long-edge graphs</a>, Bulletin of the Brazilian Mathematical Society, New Series 45.4 (2014): 625-647. Also arXiv:1303.5308 [math.AG], 2013 (see first page).
%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 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = 3*(n-1)*(n-2)*(3*n^2-3*n-11)/2.
%F a(1)=0, a(2)=0, a(3)=21, a(4)=225, a(5)=882, a(n) = 5*a(n-1)-10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - _Harvey P. Dale_, Feb 01 2013
%F G.f.: 3*x^2*(-7-40*x+11*x^2) / (x-1)^5 . - _R. J. Mathar_, Dec 19 2013
%t Table[3(n-1)(n-2)(3n^2-3n-11)/2,{n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{0,0,21,225,882},40] (* _Harvey P. Dale_, Feb 01 2013 *)
%o (PARI) concat([0,0], Vec(3*x^3*(7 + 40*x - 11*x^2) / (1 - x)^5 + O(x^40))) \\ _Colin Barker_, Nov 01 2019
%Y Severi degrees N(n, delta) for other values of delta: A033428(n-1) (1), A171113 (3), A328551 (4), A328552 (5), A171116 (6).
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_, Sep 27 2010
%E New name from _Andrey Zabolotskiy_, Jan 18 2021
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