cp's OEIS Frontend

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

%I A006849 M1947 #35 Dec 17 2021 11:27:26
%S A006849 2,9,69,567,5112,48114,469179,4691115,47849940,495893502,5206886874,
%T A006849 55273052646,592211326464,6395881806180,69555215111319,
%U A006849 761015877850035,8371174661041500,92523509359662150,1027010953940099238
%N A006849 Number of strongly self-dual planar maps with 2n edges.
%C A006849 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
%D A006849 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006849 Gheorghe Coserea, <a href="/A006849/b006849.txt">Table of n, a(n) for n = 1..200</a>
%H A006849 V. A. Liskovets, <a href="https://www.researchgate.net/publication/277008787_Enumeration_of_nonisomorphic_planar_maps">Enumeration of nonisomorphic planar maps</a>, Selecta Math. Sovietica, 4 (No. 4, 1985), 303-323.
%F A006849 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
%F A006849 G.f.: -1/2 + 1/(1 + (1 - 12*x)^(1/2)) + x/(1 + (1 - 12*x^2)^(1/2)). - _Gheorghe Coserea_, Aug 15 2015
%t A006849 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 *)
%t A006849 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 *)
%o A006849 (PARI) C = n -> binomial(2*n, n) / (n + 1);
%o A006849 a(n) = if (n%2, ( 3^n*C(n) + 3^((n-1)/2)*C((n-1)/2) )/2, 3^n*C(n)/2);
%o A006849 apply(n -> a(n), vector(30, i, i)) \\ _Gheorghe Coserea_, Aug 04 2015
%o A006849 (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
%Y A006849 Cf. A005159, A000108, A005470.
%K A006849 nonn,nice
%O A006849 1,1
%A A006849 _N. J. A. Sloane_
%E A006849 More terms from _Valery A. Liskovets_, May 27 2006