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A006849
Number of strongly self-dual planar maps with 2n edges.
(Formerly M1947)
2
2, 9, 69, 567, 5112, 48114, 469179, 4691115, 47849940, 495893502, 5206886874, 55273052646, 592211326464, 6395881806180, 69555215111319, 761015877850035, 8371174661041500, 92523509359662150, 1027010953940099238
OFFSET
1,1
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
V. A. Liskovets, Enumeration of nonisomorphic planar maps, Selecta Math. Sovietica, 4 (No. 4, 1985), 303-323.
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,nice
EXTENSIONS
More terms from Valery A. Liskovets, May 27 2006
STATUS
approved