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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).
8
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
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
KEYWORD
easy,nice,nonn
AUTHOR
Valery A. Liskovets, Apr 07 2002
STATUS
approved