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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

Table of n, a(n) for n=1..20.

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

FORMULA

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-Fran├žois Alcover, Sep 01 2019 *)

CROSSREFS

Cf. A027836, A103941, A000260.

Sequence in context: A149024 A149025 A133222 * A030928 A030912 A030892

Adjacent sequences:  A103939 A103940 A103941 * A103943 A103944 A103945

KEYWORD

easy,nonn

AUTHOR

Valery A. Liskovets, Mar 17 2005

STATUS

approved

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Last modified January 22 07:47 EST 2021. Contains 340360 sequences. (Running on oeis4.)