A090375 Number of unrooted Eulerian maps with bicolored faces which are self-isomorphic under reversing the colors.

1, 1, 2, 4, 8, 17, 40, 93, 224, 538, 1344, 3352, 8448, 21573, 54912, 143037, 366080, 968083, 2489344, 6664856, 17199104, 46515759, 120393728, 328382874, 852017152, 2340706462, 6085836800, 16822999572, 43818024960, 121777594508, 317680680960, 887053276477

Offset: 1

Author

Valery A. Liskovets, Dec 01 2003

Keywords

• easy
• nonn

Links

• M. Deryagina, On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices, Preprint 2016.
• V. Liskovets, Some easily derivable integer sequences, J. Integer Seq., v.3 (2000), Article 00.2.2.
• V. A. Liskovets, Enumerative formulas for unrooted planar maps: a pattern, Electron. J. Combin., 11:1 (2004), R88.

Crossrefs

Cf. A052701, A069727, A090371.

Programs

Mathematica

A069727[n_] := (1/(2n)) (3*2^(n - 1) Binomial[2 n, n]/((n + 1) (n + 2)) + Sum[EulerPhi[n/k] d[n/k] 2^(k - 2) Binomial[2 k, k], {k, Most[Divisors[n]]}]) + q[n]; A069727[0] = 1;
q[n_?EvenQ] := 2^((n - 4)/2) Binomial[n, n/2]/(n + 2); q[n_?OddQ] := 2^((n - 1)/2) Binomial[(n - 1), (n - 1)/2]/(n + 1);
d[n_] := 4 - Mod[n, 2];
h0[n_] := 3*2^(n - 1) Binomial[2n, n]/((n + 1)(n + 2));
A090371[n_] := (h0[n] + DivisorSum[n, If[# > 1, EulerPhi[#]*Binomial[n/# + 2, 2] h0[n/#], 0] &])/n;
a[n_] := 2 A069727[n] - A090371[n];
Array[a, 32] (* Jean-François Alcover, Aug 28 2019 *)

PARI

h0(n) = 3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2));
a090371(n) = (h0(n) + sumdiv(n, d, (d>1)*eulerphi(d)*binomial(n/d+2, 2)*h0(n/d)))/n;
d(n) = if (n%2, 3, 4);
q(n) = if (n%2, 2^((n-1)/2)*binomial(n-1, (n-1)/2)/(n+1), 2^((n-4)/2)*binomial(n, n/2)/(n+2));
a069727(n) = if (n==0, 1, q(n) + (3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2)) + sumdiv(n, k, (k!=n)*eulerphi(n/k)*d(n/k)*2^(k-2)*binomial(2*k, k)))/(2*n));
a(n) = 2*a069727(n) - a090371(n); \\ Michel Marcus, Dec 11 2014

Formula

a(n) = 2*A069727(n) - A090371(n).

a(2k+1) = 2^k*Catalan(k) = A052701(k+1).

Extensions

More terms from Michel Marcus, Dec 11 2014