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A000087
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Number of unrooted nonseparable planar maps with n edges and a distinguished face.
(Formerly M1240 N0474)
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11
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2, 1, 2, 4, 10, 37, 138, 628, 2972, 14903, 76994, 409594, 2222628, 12281570, 68864086, 391120036, 2246122574, 13025721601, 76194378042, 449155863868, 2666126033850, 15925105028685, 95664343622234, 577651490729530
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OFFSET
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1,1
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COMMENTS
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The number of unrooted non-separable n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005
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REFERENCES
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V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = (1/3n)[(n+2)binomial(3n, n)/((3n-2)(3n-1)) + Sum_{0<k<n, k|n}phi(n/k)binomial(3k, k)]+q(n) where phi is the Euler function A000010, q(n)=0 if n is even and q(n)=2(n+1)binomial(3(n+1)/2, (n+1)/2)/(3(3n-1)(3n+1)) if n is odd. - Valery A. Liskovets, Mar 17 2005
a(n) ~ 3/(8 * sqrt(3*Pi))*(27/4)^n / n^(5/2). - Cedric Lorand, Apr 18 2022
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MATHEMATICA
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q[n_] := If[EvenQ[n], 0, 2(n+1)Binomial[3(n+1)/2, (n+1)/2]/(3(3n-1)(3n+1)) ]; a[n_] := (1/(3n))((n+2)Binomial[3n, n]/((3n-2)(3n-1)) + Sum[EulerPhi[ n/k] Binomial[3k, k], {k, Divisors[n] // Most}]) + q[n]; Array[a, 30] (* Jean-François Alcover, Feb 04 2016, after Valery A. Liskovets *)
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PROG
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(PARI) q(n) = if(n%2, 2*(n + 1)*binomial(3*(n + 1)/2, (n + 1)/2) / (3*(3*n - 1)*(3*n + 1)), 0);
a(n) = (1/(3*n)) * ((n + 2) * binomial(3*n, n)/((3*n - 2) * (3*n - 1)) + sum(k=1, n - 1, if(Mod(n, k)==0, eulerphi(n/k) * binomial(3*k, k)))) + q(n); \\ Indranil Ghosh, Apr 04 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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