A103939 Number of unrooted Eulerian n-edge maps in the plane (planar with a distinguished outside face).

1, 1, 3, 8, 32, 136, 722, 3924, 22954, 138316, 860364, 5472444, 35503288, 234070648, 1564945158, 10589356592, 72412611194, 499788291616, 3478059566250, 24383023246284, 172074483068320, 1221654305104920, 8720583728414354, 62560709120463028, 450854177292364660

Offset

0,3

References

V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Formula

For n > 0, a(n) = (1/(2n))*(2^n*binomial(2n, n)/(n+1) + Sum_{0<k<n, k|n} phi(n/k)*2^k*binomial(2k, k)) where phi is the Euler function A000010.

Mathematica

a[n_] := (1/(2n)) (2^n Binomial[2n, n]/(n+1) + Sum[Boole[0<k<n] EulerPhi[ n/k] 2^k Binomial[2k, k], {k, Divisors[n]}]);
Array[a, 20] (* Jean-François Alcover, Aug 28 2019 *)

Prog

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

Crossrefs

Cf. A052701, A103940, A069727.

Sequence in context: A148915 A151541 A022563 * A094610 A064316 A009438

Adjacent sequences: A103936 A103937 A103938 * A103940 A103941 A103942

Keyword

easy,nonn

Author

Valery A. Liskovets, Mar 17 2005

Extensions

a(0)=1 prepended and terms a(21) and beyond from Andrew Howroyd, Mar 29 2021

Status

approved