A118447 Number of rooted n-edge one-vertex maps on the Klein bottle (dually: one-face maps).

4, 42, 304, 1870, 10488, 55412, 280768, 1379286, 6616360, 31144300, 144367584, 660746892, 2991902704, 13424189160, 59758420736, 264191654758, 1160934273288, 5074150057916, 22071747625120, 95596117130724

Offset: 2

Author

Valery A. Liskovets, May 04 2006

Keywords

• nonn

Comments

One-vertex maps on the projective plane are counted by A000346 and one-vertex maps on a non-orientable genus-3 surface by A118448. 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.

Programs

Mathematica

((R - 1)^2 (R + 1) (R + 3)/(8 R^5) /. R -> Sqrt[1 - 4x]) + O[x]^22 // CoefficientList[#, x]& // Drop[#, 2]& (* Jean-François Alcover, Aug 28 2019 *)

Formula

O.g.f.: (R-1)^2(R+1)(R+3)/8R^5, where R=sqrt(1-4x).

Conjecture: -(n-2)*(n-1)^2*a(n) +2*n*(4*n-5)*(n-2)*a(n-1) -8*n*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jun 22 2016

a(n) ~ n^(3/2) * 2^(2*n-1) / sqrt(Pi) * (1 - sqrt(Pi/n)/2). - Vaclav Kotesovec, Aug 28 2019