A000087 Number of unrooted nonseparable planar maps with n edges and a distinguished face.

2, 1, 2, 4, 10, 37, 138, 628, 2972, 14903, 76994, 409594, 2222628, 12281570, 68864086, 391120036, 2246122574, 13025721601, 76194378042, 449155863868, 2666126033850, 15925105028685, 95664343622234, 577651490729530

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

The number of unrooted non-separable n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005

References

• 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).

Links

• T. D. Noe, Table of n, a(n) for n=1..200
• W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.
• W. G. Brown, Enumeration of non-separable planar maps
• V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Crossrefs

Row sums of A046653.

Cf. A006402, A103938.

Programs

Mathematica

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 *)

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

Formula

a(n) = (1/3n)[(n+2)binomial(3n, n)/((3n-2)(3n-1)) + Sum_{0A000010, 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

Extensions

More terms from T. D. Noe, Mar 14 2007

Name corrected by Cyril Banderier, Apr 04 2017

Name clarified by Andrew Howroyd, Mar 29 2021