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A113183
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Number of unrooted two-face maps in the plane (considered up to orientation-preserving homeomorphism) with the faces of equal degree n: planar maps with a distinguished outside face.
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0
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1, 1, 2, 3, 8, 18, 58, 155, 546, 1592, 5774, 17798, 65676, 210362, 785248, 2588155, 9743348, 32832290, 124416022, 426685544, 1625465732, 5654938190, 21636274202, 76171463926, 292498386900, 1040120036300, 4006388161846, 14369121494126
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) = (1/n) Sum_{k|n} phi(k) C((n/k)-1,floor(n/(2k)))^2 where phi(k) is the Euler function A000010.
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EXAMPLE
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There exist 2 maps in the plane with two triangular faces: a triangle and a map consisting of a 2-path and a loop in its middle vertex that separates both ends. Therefore a(3) = 2.
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MATHEMATICA
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a[n_] := DivisorSum[n, EulerPhi[#] * Binomial[n/# - 1, Floor[n/(2*#)]]^2 &] / n; Array[a, 30] (* Amiram Eldar, Aug 24 2023 *)
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PROG
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(PARI) a(n) = sumdiv(n, k, eulerphi(k)*binomial(n/k - 1, n\(2*k))^2)/n; \\ Michel Marcus, Oct 14 2015
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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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