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A103942 Number of unrooted n-edge isthmusless maps in the plane (planar with a distinguished outside face). 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 7 06:02 EST 2023. Contains 360112 sequences. (Running on oeis4.)