A103942 Number of unrooted n-edge isthmusless maps in the plane (planar with a distinguished outside face).

1, 1, 3, 9, 38, 187, 1120, 7083, 47990, 337676, 2455517, 18310155, 139447034, 1080773098, 8502896424, 67763884363, 546147639926, 4445389286380, 36501274080076, 302060508150976, 2517213486505592, 21110062391001119, 178052027949519768, 1509631210682469661, 12860805940582898474

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))*[(5n^2+13n+2)*binomial(4n, n)/((n+1)(3n+1)(3n+2)) + Sum_{0<k<n, k|n} phi(n/k)*binomial(4k, k)+q(n)] where phi is the Euler function (A000010), q(n)=0 if n is even and q(n)=(n-1)*binomial(2n, (n-1)/2)/(n+1) if n is odd.

Mathematica

a[n_] := (1/(2n)) ((5n^2 + 13n + 2) Binomial[4n, n]/((n+1)(3n+1)(3n+2)) + Sum[Boole[0 < k < n] EulerPhi[n/k] Binomial[4k, k], {k, Divisors[n]}] + q[n]);
q[n_] := If[EvenQ[n], 0, (n-1) Binomial[2n, (n-1)/2]]/(n+1);
Array[a, 20] (* Jean-François Alcover, Sep 01 2019 *)

Prog

(PARI) a(n) = {if(n==0, 1, (sumdiv(n, d, if(d<n, 1, (5*n^2+13*n+2)/((n+1)*(3*n+1)*(3*n+2))) * eulerphi(n/d) * binomial(4*d, d)) + if(n%2, (n-1)*binomial(2*n, (n-1)/2)/(n+1)))/(2*n))} \\ Andrew Howroyd, Mar 28 2021

Crossrefs

Cf. A027836, A006390, A103941, A000260.

Sequence in context: A351070 A346056 A133222 * A030928 A030912 A030892

Adjacent sequences: A103939 A103940 A103941 * A103943 A103944 A103945

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 28 2021

Status

approved