cp's OEIS Frontend

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

%I A090375 #22 Aug 28 2019 10:35:28
%S A090375 1,1,2,4,8,17,40,93,224,538,1344,3352,8448,21573,54912,143037,366080,
%T A090375 968083,2489344,6664856,17199104,46515759,120393728,328382874,
%U A090375 852017152,2340706462,6085836800,16822999572,43818024960,121777594508,317680680960,887053276477
%N A090375 Number of unrooted Eulerian maps with bicolored faces which are self-isomorphic under reversing the colors.
%H A090375 M. Deryagina, <a href="ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v446/p031.pdf">On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices</a>, Preprint 2016.
%H A090375 V. Liskovets, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/LISK/Derseq.html">Some easily derivable integer sequences</a>, J. Integer Seq., v.3 (2000), Article 00.2.2.
%H A090375 V. A. Liskovets, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i1r88">Enumerative formulas for unrooted planar maps: a pattern</a>, Electron. J. Combin., 11:1 (2004), R88.
%F A090375 a(n) = 2*A069727(n) - A090371(n).
%F A090375 a(2k+1) = 2^k*Catalan(k) = A052701(k+1).
%t A090375 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;
%t A090375 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);
%t A090375 d[n_] := 4 - Mod[n, 2];
%t A090375 h0[n_] := 3*2^(n - 1) Binomial[2n, n]/((n + 1)(n + 2));
%t A090375 A090371[n_] := (h0[n] + DivisorSum[n, If[# > 1, EulerPhi[#]*Binomial[n/# + 2, 2] h0[n/#], 0] &])/n;
%t A090375 a[n_] := 2 A069727[n] - A090371[n];
%t A090375 Array[a, 32] (* _Jean-François Alcover_, Aug 28 2019 *)
%o A090375 (PARI) h0(n) = 3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2));
%o A090375 a090371(n) = (h0(n) + sumdiv(n, d, (d>1)*eulerphi(d)*binomial(n/d+2, 2)*h0(n/d)))/n;
%o A090375 d(n) = if (n%2, 3, 4);
%o A090375 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));
%o A090375 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));
%o A090375 a(n) = 2*a069727(n) - a090371(n); \\ _Michel Marcus_, Dec 11 2014
%Y A090375 Cf. A052701, A069727, A090371.
%K A090375 easy,nonn
%O A090375 1,3
%A A090375 _Valery A. Liskovets_, Dec 01 2003
%E A090375 More terms from _Michel Marcus_, Dec 11 2014