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A090372
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Number of unrooted planar 3-constellations with n triangles.
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3
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1, 6, 22, 174, 1479, 16808, 201834, 2631594, 35965555, 512062566, 7528425420, 113708935808, 1756853846316, 27676951028496, 443411345677658, 7209139541742750, 118738765611199983, 1978360119497335826
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OFFSET
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1,2
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COMMENTS
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These are planar maps with bicolored faces having n black triangular faces and an arbitrary number of white faces of degrees multiple to 3. The vertices can be and are colored so that any black triangle is colored counterclockwise 1,2,3. Isomorphisms are required to respect the colorings. Also unrooted bi-Eulerian maps with bicolored both vertices and faces and with 2n edges; the maps are considered up to color-preserve isomorphism.
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LINKS
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MAPLE
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with(numtheory): C_3 := proc(n) local s, d; if n=0 then RETURN(1) else s := -3^n*binomial(3*n, n); for d in divisors(n) do s := s+phi(n/d)*3^d*binomial(3*d, d) od; RETURN((4/(3*n))*(3^n*binomial(3*n, n)/((2*n+1)*(2*n+2))+s/2)); fi; end;
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MATHEMATICA
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a[0] = 1; a[n_] := Module[{s, d}, s = -3^n Binomial[3n, n]; Do[s = s + EulerPhi[n/d] 3^d Binomial[3d, d], {d, Divisors[n]}]; (4/(3n)) (3^n Binomial[3n, n]/((2n+1)(2n+2)) + s/2)];
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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