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A118447
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Number of rooted n-edge one-vertex maps on the Klein bottle (dually: one-face maps).
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1
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4, 42, 304, 1870, 10488, 55412, 280768, 1379286, 6616360, 31144300, 144367584, 660746892, 2991902704, 13424189160, 59758420736, 264191654758, 1160934273288, 5074150057916, 22071747625120, 95596117130724
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refs;
listen;
history;
text;
internal format)
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OFFSET
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2,1
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COMMENTS
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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).
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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MATHEMATICA
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((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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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