A006849 Number of strongly self-dual planar maps with 2n edges.

2, 9, 69, 567, 5112, 48114, 469179, 4691115, 47849940, 495893502, 5206886874, 55273052646, 592211326464, 6395881806180, 69555215111319, 761015877850035, 8371174661041500, 92523509359662150, 1027010953940099238

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

A planar map is called strongly self-dual if it is self-dual with respect to an orientation-preserving duality. - Valery A. Liskovets, May 27 2006

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..200
• V. A. Liskovets, Enumeration of nonisomorphic planar maps, Selecta Math. Sovietica, 4 (No. 4, 1985), 303-323.

Crossrefs

Cf. A005159, A000108, A005470.

Programs

Mathematica

With[{nn = 21}, CoefficientList[InverseSeries[Series[2*x/(12*x^2 + 12*x + 3), {x, 0, nn}]] + InverseSeries[Series[2*x/(12*x^2 + 1), {x, 0, nn}]], x]] (* Gheorghe Coserea, Aug 15 2015 *)
a[n_] := 3^n*CatalanNumber[n]/2 + If[OddQ[n], 3^((n-1)/2)*CatalanNumber[(n-1)/2]/2, 0]; Array[a, 20] (* Jean-François Alcover, Jan 17 2018 *)

PARI

C = n -> binomial(2*n, n) / (n + 1);
a(n) = if (n%2, ( 3^n*C(n) + 3^((n-1)/2)*C((n-1)/2) )/2, 3^n*C(n)/2);
apply(n -> a(n), vector(30, i, i)) \\ Gheorghe Coserea, Aug 04 2015

PARI

x='x + O('x^33); Vec(-1/2 + 1/(1 + (1 - 12*x)^(1/2)) + x/(1 + (1 - 12*x^2)^(1/2))) \\ Gheorghe Coserea, Aug 15 2015

Formula

a(2k) = 3^(2k)C(2k)/2=A005159(2k)/2 (4k edges, k>0) and a(2k-1) = (3^(2k-1)C(2k-1)+3^(k-1)C(k-1))/2 =(A005159(2k-1)+A005159(k-1))/2 (4k-2 edges, k>0) where C(n) = A000108(n) (Catalan numbers). - Valery A. Liskovets, May 27 2006

G.f.: -1/2 + 1/(1 + (1 - 12*x)^(1/2)) + x/(1 + (1 - 12*x^2)^(1/2)). - Gheorghe Coserea, Aug 15 2015

Extensions

More terms from Valery A. Liskovets, May 27 2006