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

1, 3, 7, 33, 156, 1070, 7515, 59151, 483925, 4136964, 36416865, 329048627, 3037029030, 28553451498, 272766018806, 2642420298576, 25916954091582, 257009789443925, 2573962338306141, 26008719387850068, 264933535266372732

Offset: 1

Author

N. J. A. Sloane, May 24 2000

Keywords

• nonn
• easy

Comments

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.

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..200
• V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Crossrefs

Average of A006384 and A006849, the latter interspersed with 0's (cf. formula).

Programs

Mathematica

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];
q[n_?OddQ] := 2*(3^((n - 1)/2)/(n + 1))*Binomial[ n - 1, (n - 1)/2];
q[n_?EvenQ] := 2*(n-1)*(3^((n-2)/2)/(n*(n+2)))*Binomial[ n - 2, (n - 2)/2];
a6849[n_] := 3^n*CatalanNumber[n]/2 + If[OddQ[n], 3^((n - 1)/2)* CatalanNumber[(n - 1)/2]/2, 0];
a[n_] := If[OddQ[n], a6384[n]/2, (a6384[n] + a6849[n/2])/2];
Array[a, 21] (* Jean-François Alcover, Aug 30 2019 *)

PARI

F(n) = { 3^n * binomial(2*n,n); }
S(n) = { my(acc = 0);
         fordiv(n, d, if(d != n, acc += eulerphi(n/d) * F(d)));
         return(acc); }
Q(n) = { if (n%2, 2 * F((n-1)/2) / (n+1),
                  2 * F((n-2)/2) * (n-1)/(n*(n+2))); }
A006384(n) = { if (n < 0, return(0)); if (n == 0, return(1));
               (2*F(n)/((n+1)*(n+2)) + S(n)) / (2*n) + Q(n); }
G(n) = { 3^n * binomial(2*n,n) / (n + 1); }
A006849(n) = { if (n <= 0, return(0));
               if (n%2, (G(n) + G((n-1)/2)) / 2, G(n)/2); }
a(n) = { if (n <= 0, return(0));
         if (n%2,  A006384(n)/2, (A006384(n) + A006849(n/2))/2) };
apply(n->a(n), vector(33, i, i)) \\ Gheorghe Coserea, Aug 20 2015

Formula

a(2k+1) = A006384(2k+1)/2 and a(2k) = (A006384(2k) + A006849(k))/2. - Gheorghe Coserea, Aug 05 2015

Extensions

More terms from Valery A. Liskovets, May 27 2006

More terms from Sean A. Irvine, Mar 24 2013