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A118451 Number of rooted n-edge maps on a non-orientable genus-3 surface.

%I A118451 #17 Oct 27 2024 06:00:34
%S A118451 41,1380,31225,592824,10185056,164037704,2525186319,37596421940,
%T A118451 545585129474,7758174844664,108518545261360,1497384373878512,
%U A118451 20426386710028260,275940187259609296,3696482210884173349
%N A118451 Number of rooted n-edge maps on a non-orientable genus-3 surface.
%D A118451 E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
%D A118451 D. M. Jackson and T. I. Visentin, An atlas of the smaller maps in orientable and nonorientable surfaces. CRC Press, Boca Raton, 2001.
%H A118451 R. J. Mathar, <a href="/A118451/b118451.txt">Table of n, a(n) for n = 3..100</a>
%H A118451 Didier Arquès, Alain Giorgetti, <a href="https://doi.org/10.1016/S0304-3975(98)00230-8">Counting rooted maps on a surface</a>, Theoret. Comput. Sci. 234 (2000), no. 1-2, 255--272. MR1745078 (2001f:05078). - _N. J. A. Sloane_, Jul 27 2012
%F A118451 O.g.f.: (R-1) *(R+1) *(68*R^5 +280*R^4 +588*R^3 +808*R^2 +416*R -(28*R^4+59*R^3+114*R^2+119*R+40)*sqrt(12*R*(R+2)))/ (96*R^5*(R+2)^3), where R=sqrt(1-12*x).
%F A118451 a(n) ~ 2^(2*n + 1/2) * 3^(n - 1/2) * n^(5/4) / Gamma(1/4) * (1 - 13*Gamma(1/4) / (8*sqrt(6)*n^(1/4)) + 23*Gamma(1/4)^2 / (32*Pi*sqrt(2*n)) - 23*Gamma(1/4) / (16*sqrt(6*Pi)*n^(3/4))). - _Vaclav Kotesovec_, Oct 27 2024
%p A118451 R := sqrt(1-12*x) ;
%p A118451 (R-1)*(R+1)*(68*R^5+280*R^4+588*R^3+808*R^2+416*R -(28*R^4+59*R^3+114*R^2+119*R+40)*sqrt(12*R*(R+2)))/96/R^5/(R+2)^3 ;
%p A118451 g := series(%,x=0,101) ;
%p A118451 for n from 3 to 100 do
%p A118451     printf("%d %d\n",n,coeftayl(g,x=0,n)) ;
%p A118451 end do: # _R. J. Mathar_, Oct 17 2012
%t A118451 R = Sqrt[1-12x];
%t A118451 (R-1)(R+1)(68R^5 + 280R^4 + 588R^3 + 808R^2 + 416R - (28R^4 + 59R^3 + 114R^2 + 119R + 40) Sqrt[12R(R+2)])/96/R^5/(R+2)^3 + O[x]^18 // CoefficientList[#, x]& // Drop[#, 3]& (* _Jean-François Alcover_, Aug 28 2019 *)
%Y A118451 Cf. A006344, A214334.
%K A118451 nonn
%O A118451 3,1
%A A118451 _Valery A. Liskovets_, May 04 2006