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

41, 1380, 31225, 592824, 10185056, 164037704, 2525186319, 37596421940, 545585129474, 7758174844664, 108518545261360, 1497384373878512, 20426386710028260, 275940187259609296, 3696482210884173349

Offset: 3

Author

Valery A. Liskovets, May 04 2006

Keywords

• nonn

References

• E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
• D. M. Jackson and T. I. Visentin, An atlas of the smaller maps in orientable and nonorientable surfaces. CRC Press, Boca Raton, 2001.

Links

• R. J. Mathar, Table of n, a(n) for n = 3..100
• Didier Arquès, Alain Giorgetti, Counting rooted maps on a surface, Theoret. Comput. Sci. 234 (2000), no. 1-2, 255--272. MR1745078 (2001f:05078). - _N. J. A. Sloane_, Jul 27 2012

Crossrefs

Cf. A006344, A214334.

Programs

Maple

R := sqrt(1-12*x) ;
(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 ;
g := series(%,x=0,101) ;
for n from 3 to 100 do
    printf("%d %d\n",n,coeftayl(g,x=0,n)) ;
end do: # R. J. Mathar, Oct 17 2012

Mathematica

R = Sqrt[1-12x];
(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 *)

Formula

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).

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