login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000087 Number of unrooted planar maps.
(Formerly M1240 N0474)
10

%I M1240 N0474

%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 planar maps.

%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

%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-Fran├žois 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 Cf. A103938, row sums of A046653.

%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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 28 11:40 EDT 2020. Contains 334681 sequences. (Running on oeis4.)