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%I M1240 N0474 #55 Apr 22 2022 18:07:44
%S 2,1,2,4,10,37,138,628,2972,14903,76994,409594,2222628,12281570,
%T 68864086,391120036,2246122574,13025721601,76194378042,449155863868,
%U 2666126033850,15925105028685,95664343622234,577651490729530
%N Number of unrooted nonseparable planar maps with n edges and a distinguished face.
%C The number of unrooted non-separable n-edge maps in the plane (planar with a distinguished outside face). - _Valery A. Liskovets_, Mar 17 2005
%D V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000087/b000087.txt">Table of n, a(n) for n=1..200</a>
%H W. G. Brown, <a href="http://dx.doi.org/10.4153/CJM-1963-056-7">Enumeration of non-separable planar maps</a>, Canad. J. Math., 15 (1963), 526-545.
%H W. G. Brown, <a href="/A000087/a000087.pdf">Enumeration of non-separable planar maps</a>
%H V. A. Liskovets and T. R. Walsh, <a href="http://dx.doi.org/10.1016/j.aam.2005.03.006">Counting unrooted maps on the plane</a>, Advances in Applied Math., 36, No.4 (2006), 364-387.
%F 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
%F a(n) ~ 3/(8 * sqrt(3*Pi))*(27/4)^n / n^(5/2). - _Cedric Lorand_, Apr 18 2022
%t 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_ *)
%o (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);
%o 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
%Y Row sums of A046653.
%Y Cf. A006402, A103938.
%K nonn
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _T. D. Noe_, Mar 14 2007
%E Name corrected by _Cyril Banderier_, Apr 04 2017
%E Name clarified by _Andrew Howroyd_, Mar 29 2021
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