A069727 Number of nonisomorphic unrooted Eulerian planar maps with n edges (Eulerian means that all vertices are of even valency; there is an Eulerian cycle).

1, 1, 2, 4, 12, 34, 154, 675, 3534, 18985, 108070, 632109, 3807254, 23411290, 146734695, 934382820, 6034524474, 39457153432, 260855420489, 1741645762265, 11732357675908, 79673115468562, 545036528857605, 3753642607424647, 26010818244754788, 181266500331748878

Offset

0,3

Comments

By duality, also the number of unrooted (sensed) bipartite maps with n edges. - Andrew Howroyd, Mar 29 2021

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

M. Deryagina, On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices, Preprint 2016.

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Formula

a(n) = (1/(2n))*(3*2^(n-1)*binomial(2n, n)/((n+1)(n+2)) + Sum_{k=1..n-1, k|n} phi(n/k)*d(n/k)*2^(k-2)*binomial(2k, k)) + q(n) where phi is the Euler function A000010, q(n) = 2^((n-4)/2)*binomial(n, n/2)/(n+2) if n is even, q(n) = 2^((n-1)/2)*binomial(n-1, (n-1)/2)/(n+1) if n is odd, d(n)=4, if n is even and d(n)=3 if n is odd. - Valery A. Liskovets, Mar 17 2005

a(n) ~ 3 * 2^(3*n-2) / (sqrt(Pi) * n^(7/2)). - Vaclav Kotesovec, Aug 28 2019

Mathematica

a[n_] := (1/(2n)) * (3*2^(n-1) * Binomial[2n, n]/((n+1)*(n+2)) + Sum[ EulerPhi[n/k] * d[n/k] * 2^(k-2) * Binomial[2k, k], {k, Most[ Divisors[n]]}]) + q[n]; a[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]; Table[ a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 19 2011, after Valery A. Liskovets *)

Prog

(PARI) a(n) = {if(n==0, 1, sumdiv(n, d, if(d<n, 1, 2/((n+1)*(n+2))) * eulerphi(n/d) * (4-n/d%2) * 2^(d-2) * binomial(2*d, d))/(2*n) + if(n%2, 2^((n-1)/2)/(n+1), 2^((n-4)/2)/(n+2))*binomial(n\2*2, n\2))} \\ Andrew Howroyd, Mar 29 2021

Crossrefs

Cf. A000257 (rooted), A069720, A069724, A103939 (with distinguished face), A103940 (with distinguished vertex).

Sequence in context: A363202 A215953 A209027 * A148204 A151525 A148205

Adjacent sequences: A069724 A069725 A069726 * A069728 A069729 A069730

Keyword

easy,nice,nonn

Author

Valery A. Liskovets, Apr 07 2002

Status

approved