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A318104
Number of genus 4 rooted hypermaps with n darts.
3
8064, 579744, 23235300, 684173164, 16497874380, 344901105444, 6471056247920, 111480953909328, 1792031518697232, 27197316623478960, 393207192141924744, 5453210050430783640, 72949244341257096792, 945523594111460363208, 11918067649004916470640, 146538779626167833263888, 1762112462707129510538640
OFFSET
9,1
COMMENTS
Column k = 4 of A321710.
a(n) = 0 for n < 9. - N. J. A. Sloane, Dec 24 2018
LINKS
Mednykh, A.; Nedela, R. Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 6
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.
FORMULA
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.
EXAMPLE
A(x) = 8064*x^9 + 579744*x^10 + 23235300*x^11 + 684173164*x^12 + ...
MATHEMATICA
y = (1 - Sqrt[1 - 8 x])/(4 x);
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);
Drop[CoefficientList[gf + O[x]^26, x], 9] (* Jean-François Alcover, Feb 07 2019, from PARI *)
PROG
(PARI)
seq(N) = {
my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
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));
};
seq(17)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Nov 12 2018
STATUS
approved