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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 (list; graph; refs; listen; history; text; internal format)
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

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.

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-Fran├žois 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

Cf. A005159, A000108, A005470.

Sequence in context: A193160 A255537 A272663 * A319285 A316652 A121417

Adjacent sequences:  A006846 A006847 A006848 * A006850 A006851 A006852

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Valery A. Liskovets, May 27 2006

STATUS

approved

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Last modified March 26 04:32 EDT 2019. Contains 321481 sequences. (Running on oeis4.)