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A054935 Number of planar maps with n edges up to orientation-preserving duality.

%I A054935 #40 Aug 30 2019 08:30:53
%S A054935 1,3,7,33,156,1070,7515,59151,483925,4136964,36416865,329048627,
%T A054935 3037029030,28553451498,272766018806,2642420298576,25916954091582,
%U A054935 257009789443925,2573962338306141,26008719387850068,264933535266372732
%N A054935 Number of planar maps with n edges up to orientation-preserving duality.
%C A054935 Replacing each edge by a vertex of degree 4, one sees that a(n) is also the number of non-isomorphic planar maps (a.k.a. clean dessins on the Riemann sphere) with n vertices of degree 4, and 2n edges.
%H A054935 Gheorghe Coserea, <a href="/A054935/b054935.txt">Table of n, a(n) for n = 1..200</a>
%H A054935 V. A. Liskovets, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LISK/Derseq.html">Some easily derivable sequences</a>, J. Integer Sequences, 3 (2000), #00.2.2.
%F A054935 a(2k+1) = A006384(2k+1)/2 and a(2k) = (A006384(2k) + A006849(k))/2. - _Gheorghe Coserea_, Aug 05 2015
%t A054935 a6384[0] = 1; a6384[n_] := (1/(2n))*(2*(3^n/((n + 1)*(n + 2)))*Binomial[2 n, n] + Sum[ EulerPhi[n/k]*3^k*Binomial[ 2k, k], {k, Most[ Divisors[ n]]}]) + q[n];
%t A054935 q[n_?OddQ] := 2*(3^((n - 1)/2)/(n + 1))*Binomial[ n - 1, (n - 1)/2];
%t A054935 q[n_?EvenQ] := 2*(n-1)*(3^((n-2)/2)/(n*(n+2)))*Binomial[ n - 2, (n - 2)/2];
%t A054935 a6849[n_] := 3^n*CatalanNumber[n]/2 + If[OddQ[n], 3^((n - 1)/2)* CatalanNumber[(n - 1)/2]/2, 0];
%t A054935 a[n_] := If[OddQ[n], a6384[n]/2, (a6384[n] + a6849[n/2])/2];
%t A054935 Array[a, 21] (* _Jean-François Alcover_, Aug 30 2019 *)
%o A054935 (PARI)
%o A054935 F(n) = { 3^n * binomial(2*n,n); }
%o A054935 S(n) = { my(acc = 0);
%o A054935          fordiv(n, d, if(d != n, acc += eulerphi(n/d) * F(d)));
%o A054935          return(acc); }
%o A054935 Q(n) = { if (n%2, 2 * F((n-1)/2) / (n+1),
%o A054935                   2 * F((n-2)/2) * (n-1)/(n*(n+2))); }
%o A054935 A006384(n) = { if (n < 0, return(0)); if (n == 0, return(1));
%o A054935                (2*F(n)/((n+1)*(n+2)) + S(n)) / (2*n) + Q(n); }
%o A054935 G(n) = { 3^n * binomial(2*n,n) / (n + 1); }
%o A054935 A006849(n) = { if (n <= 0, return(0));
%o A054935                if (n%2, (G(n) + G((n-1)/2)) / 2, G(n)/2); }
%o A054935 a(n) = { if (n <= 0, return(0));
%o A054935          if (n%2,  A006384(n)/2, (A006384(n) + A006849(n/2))/2) };
%o A054935 apply(n->a(n), vector(33, i, i)) \\ _Gheorghe Coserea_, Aug 20 2015
%Y A054935 Average of A006384 and A006849, the latter interspersed with 0's (cf. formula).
%K A054935 nonn,easy
%O A054935 1,2
%A A054935 _N. J. A. Sloane_, May 24 2000
%E A054935 More terms from _Valery A. Liskovets_, May 27 2006
%E A054935 More terms from _Sean A. Irvine_, Mar 24 2013