A103942 - OEIS

%I #18 Mar 28 2021 19:31:23

%S 1,1,3,9,38,187,1120,7083,47990,337676,2455517,18310155,139447034,

%T 1080773098,8502896424,67763884363,546147639926,4445389286380,

%U 36501274080076,302060508150976,2517213486505592,21110062391001119,178052027949519768,1509631210682469661,12860805940582898474

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

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

%H Andrew Howroyd, <a href="/A103942/b103942.txt">Table of n, a(n) for n = 0..500</a>

%H V. A. Liskovets and T. R. Walsh, <a href="http://dx.doi.org/10.1016/j.aam.2005.03.006">Counting unrooted maps on the plane</a>, Advances in Applied Math., 36, No.4 (2006), 364-387.

%F 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.

%t 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]);

%t q[n_] := If[EvenQ[n], 0, (n-1) Binomial[2n, (n-1)/2]]/(n+1);

%t Array[a, 20] (* _Jean-François Alcover_, Sep 01 2019 *)

%o (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

%Y Cf. A027836, A006390, A103941, A000260.

%K easy,nonn

%O 0,3

%A _Valery A. Liskovets_, Mar 17 2005

%E a(0)=1 prepended and terms a(21) and beyond from _Andrew Howroyd_, Mar 28 2021