A118448 Number of rooted n-edge one-vertex maps on a non-orientable genus-3 surface (dually: one-face maps).

41, 690, 7150, 58760, 420182, 2736524, 16661580, 96411060, 536075430, 2886649260, 15139322276, 77665981120, 391031449340, 1937266785080, 9464122525784, 45670084085004, 218002466412870, 1030588793671980

Offset: 3

Author

Valery A. Liskovets, May 04 2006

Keywords

• nonn

Comments

One-vertex maps on the Klein bottle are counted by A118447 and one-vertex maps on a non-orientable genus-4 surface by A118449. Such maps are also called bouquets of loops (and their duals are called unicellular maps).

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

A diagonal of A214337.

Programs

Mathematica

((R-1)^3 (R+1)^2 (11 R^2 - 29 R - 64)/(64 R^8) /. R -> Sqrt[1-4x]) + O[x]^21 // CoefficientList[#, x]& // Drop[#, 3]& (* Jean-François Alcover, Aug 29 2019 *)

Formula

O.g.f.: (R-1)^3*(R+1)^2*(11*R^2-29*R-64)/(64*R^8), where R=sqrt(1-4*x).

D-finite with recurrence (69104*n+95905)*(n-2)*(n-3) *a(n) +2*(n-3) *(34552*n^2-2691825*n+3948578) *a(n-1) +4*(-967456*n^3+10134720*n^2-23520179*n+15213000) *a(n-2) + 144 *(2*n-5) *(34552*n-41477) *(n-2) *a(n-3)=0. R. J. Mathar, Oct 17 2012

a(n) ~ n^3 * 2^(2*n-1) / 3 * (1 - 7/(4*sqrt(Pi*n))). - Vaclav Kotesovec, Oct 27 2024