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

1,3

Links

Table of n, a(n) for n=1..32.

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.

Formula

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

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

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 *)

Prog

(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

Crossrefs

Cf. A052701, A069727, A090371.

Sequence in context: A137856 A373353 A304970 * A210540 A332398 A307555

Adjacent sequences: A090372 A090373 A090374 * A090376 A090377 A090378

Keyword

easy,nonn

Author

Valery A. Liskovets, Dec 01 2003

Extensions

More terms from Michel Marcus, Dec 11 2014

Status

approved