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%I #20 Aug 23 2022 14:06:00
%S 8064,579744,23235300,684173164,16497874380,344901105444,
%T 6471056247920,111480953909328,1792031518697232,27197316623478960,
%U 393207192141924744,5453210050430783640,72949244341257096792,945523594111460363208,11918067649004916470640,146538779626167833263888,1762112462707129510538640
%N Number of genus 4 rooted hypermaps with n darts.
%C Column k = 4 of A321710.
%C a(n) = 0 for n < 9. - _N. J. A. Sloane_, Dec 24 2018
%H Gheorghe Coserea, <a href="/A318104/b318104.txt">Table of n, a(n) for n = 9..109</a>
%H Mednykh, A.; Nedela, R. <a href="https://doi.org/10.1007/s10958-017-3555-5">Recent progress in enumeration of hypermaps</a>, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 6
%H Timothy R. Walsh, <a href="http://www.info2.uqam.ca/~walsh_t/papers/GENERATING NONISOMORPHIC.pdf">Space-efficient generation of nonisomorphic maps and hypermaps</a>
%H T. R. Walsh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Walsh/walsh3.html">Space-Efficient Generation of Nonisomorphic Maps and Hypermaps</a>, J. Int. Seq. 18 (2015) # 15.4.3.
%H Peter Zograf, <a href="https://arxiv.org/abs/1312.2538">Enumeration of Grothendieck's Dessins and KP Hierarchy</a>, arXiv:1312.2538 [math.CO], 2014.
%F G.f.: -y*(y - 1)^9*(262*y^14 - 4716*y^13 + 78327*y^12 - 569134*y^11 + 3266910*y^10 - 12675726*y^9 + 37548087*y^8 - 82680972*y^7 + 137674842*y^6 - 170295272*y^5 + 152918277*y^4 - 94811622*y^3 + 37127810*y^2 - 7566846*y + 505869)/(4*(y - 2)^17*(y + 1)^13), where y = C(2*x), C being the g.f. for A000108.
%e A(x) = 8064*x^9 + 579744*x^10 + 23235300*x^11 + 684173164*x^12 + ...
%t y = (1 - Sqrt[1 - 8 x])/(4 x);
%t gf = -y (y-1)^9 (262 y^14 - 4716 y^13 + 78327 y^12 - 569134 y^11 + 3266910 y^10 - 12675726 y^9 + 37548087 y^8 - 82680972 y^7 + 137674842 y^6 - 170295272 y^5 + 152918277 y^4 - 94811622 y^3 + 37127810 y^2 - 7566846 y + 505869)/(4 (y-2)^17 (y+1)^13);
%t Drop[CoefficientList[gf + O[x]^26, x], 9] (* _Jean-François Alcover_, Feb 07 2019, from PARI *)
%o (PARI)
%o seq(N) = {
%o my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
%o Vec(-y*(y - 1)^9*(262*y^14 - 4716*y^13 + 78327*y^12 - 569134*y^11 + 3266910*y^10 - 12675726*y^9 + 37548087*y^8 - 82680972*y^7 + 137674842*y^6 - 170295272*y^5 + 152918277*y^4 - 94811622*y^3 + 37127810*y^2 - 7566846*y + 505869)/(4*(y - 2)^17*(y + 1)^13));
%o };
%o seq(17)
%Y Cf. A000257, A118093, A214817, A214818, A321710.
%K nonn
%O 9,1
%A _Gheorghe Coserea_, Nov 12 2018