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

1,2

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.

Formula

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A057480 A051256 A057795 * A208989 A358961 A024496

Adjacent sequences: A054932 A054933 A054934 * A054936 A054937 A054938

Keyword

nonn,easy

Author

N. J. A. Sloane, May 24 2000

Extensions

More terms from Valery A. Liskovets, May 27 2006

More terms from Sean A. Irvine, Mar 24 2013

Status

approved