login
This site is supported by donations to The OEIS Foundation.

 

Logo

The OEIS is looking to hire part-time people to help edit core sequences, upload scanned documents, process citations, fix broken links, etc. - Neil Sloane, njasloane@gmail.com

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000087 Number of unrooted planar maps.
(Formerly M1240 N0474)
9
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

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-Fran├žois 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

Cf. A103938.

Sequence in context: A063894 A268619 A024500 * 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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 27 07:46 EDT 2017. Contains 287198 sequences.