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A000087 Number of unrooted nonseparable planar maps with n edges and a distinguished face.
(Formerly M1240 N0474)
11
2, 1, 2, 4, 10, 37, 138, 628, 2972, 14903, 76994, 409594, 2222628, 12281570, 68864086, 391120036, 2246122574, 13025721601, 76194378042, 449155863868, 2666126033850, 15925105028685, 95664343622234, 577651490729530 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The number of unrooted non-separable n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005

REFERENCES

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=1..200

W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.

W. G. Brown, Enumeration of non-separable planar maps

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) = (1/3n)[(n+2)binomial(3n, n)/((3n-2)(3n-1)) + Sum_{0<k<n, k|n}phi(n/k)binomial(3k, k)]+q(n) where phi is the Euler function A000010, q(n)=0 if n is even and q(n)=2(n+1)binomial(3(n+1)/2, (n+1)/2)/(3(3n-1)(3n+1)) if n is odd. - Valery A. Liskovets, Mar 17 2005

a(n) ~ 3/(8 * sqrt(3*Pi))*(27/4)^n / n^(5/2). - Cedric Lorand, Apr 18 2022

MATHEMATICA

q[n_] := If[EvenQ[n], 0, 2(n+1)Binomial[3(n+1)/2, (n+1)/2]/(3(3n-1)(3n+1)) ]; a[n_] := (1/(3n))((n+2)Binomial[3n, n]/((3n-2)(3n-1)) + Sum[EulerPhi[ n/k] Binomial[3k, k], {k, Divisors[n] // Most}]) + q[n]; Array[a, 30] (* Jean-François Alcover, Feb 04 2016, after Valery A. Liskovets *)

PROG

(PARI) q(n) = if(n%2, 2*(n + 1)*binomial(3*(n + 1)/2, (n + 1)/2) / (3*(3*n - 1)*(3*n + 1)), 0);

a(n) = (1/(3*n)) * ((n + 2) * binomial(3*n, n)/((3*n - 2) * (3*n - 1)) + sum(k=1, n - 1, if(Mod(n, k)==0, eulerphi(n/k) * binomial(3*k, k)))) + q(n); \\ Indranil Ghosh, Apr 04 2017

CROSSREFS

Row sums of A046653.

Cf. A006402, A103938.

Sequence in context: A268619 A024500 A318870 * A145667 A095067 A225564

Adjacent sequences: A000084 A000085 A000086 * A000088 A000089 A000090

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from T. D. Noe, Mar 14 2007

Name corrected by Cyril Banderier, Apr 04 2017

Name clarified by Andrew Howroyd, Mar 29 2021

STATUS

approved

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Last modified December 5 06:35 EST 2022. Contains 358582 sequences. (Running on oeis4.)