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A006384 Number of planar maps with n edges.
(Formerly M1281)
1, 2, 4, 14, 57, 312, 2071, 15030, 117735, 967850, 8268816, 72833730, 658049140, 6074058060, 57106433817, 545532037612, 5284835906037, 51833908183164, 514019531037910, 5147924676612282, 52017438279806634, 529867070532745464 (list; graph; refs; listen; history; text; internal format)



V. A. Liskovets, A census of nonisomorphic planar maps, in Algebraic Methods in Graph Theory, Vol. II, ed. L. Lovasz and V. T. Sos, North-Holland, 1981.

V. A. Liskovets, Enumeration of nonisomorphic planar maps, Selecta Math. Sovietica, 4 (No. 4, 1985), 303-323.

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

T. R. S. Walsh, personal communication.


Alois P. Heinz, Table of n, a(n) for n = 0..300

V. A. Liskovets, Enumerative formulas for unrooted planar maps: a pattern, Electron. J. Combin., 11:1 (2004), R88.

Valery A. Liskovets, A reductive technique for enumerating non-isomorphic planar maps, Discrete Math. 156 (1996), no. 1-3, 197--217. MR1405018 (97f:05087) - From N. J. A. Sloane, Jun 03 2012

Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.

N. J. A. Sloane, Notes

T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices

T. R. S. Walsh, Data (Preprint 1, Part 1)

T. R. S. Walsh, Data (Preprint 1, Part 2)

T. R. S. Walsh, Data (Preprint 1, Part 3)

T. R. S. Walsh, Notes

T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices

T. R. S. Walsh & N. J. A. Sloane, Correspondence, 1991

Nicholas C. Wormald, Counting unrooted planar maps, Discrete Math. 36 (1981), no. 2, 205-225.

N. C. Wormald, On the number of planar maps, Can. J. Math. 33.1 (1981), 1-11. (Annotated scanned copy)


For n>0, a(n) = (1/2n)[A'(n)+sum_{k<n,k|n}phi(n/k) binomial(k+2,2) A'(k)]+q(n) where phi(n) is the Euler function A000010, q(n)=(n+3) A'(n-1/2)/4 if n is odd and q(n) = (n-1)A'(n-2/2)/4 if n is even, where A'(n)=A000168(n), the number of rooted maps. - Valery A. Liskovets, May 27 2006

Equivalently, a(n) = (1/2n)[2*3^n/((n+1)(n+2))*binomial(2n,n) +sum_{k<n,k|n} phi(n/k)3^k*binomial(2k,k)]+q(n) where q(n)=2*3^((n-1)/ 2)/ (n+1)*binomial(n-1,(n-1)/2) if n is odd and q(n)=2(n-1)*3^((n-2)/2)/ (n(n+2))*binomial(n-2,(n-2)/2) if n is even. - Valery A. Liskovets, May 27 2006

a(n) ~ 12^n / (sqrt(Pi) * n^(7/2)). - Vaclav Kotesovec, Sep 12 2014


with(numtheory): a:= n-> `if` (n=0, 1, floor (2*3^n /(n+1)/(n+2) *binomial(2*n, n) +add (phi(n/t) *3^t *binomial(2*t, t), t=divisors(n) minus {n}))/2/n +`if` (irem(n, 2)=1, 2*3^((n-1)/2) /(n+1) *binomial(n-1, (n-1)/2), 2*(n-1) *3^((n-2)/2) /n/(n+2) *binomial(n-2, (n-2)/2))): seq (a(n), n=0..30); # Alois P. Heinz, Apr 24 2009


a[0] = 1; a[n_] := (1/(2n))*(2*(3^n/((n+1)*(n+2)))*Binomial[2n, n] + Sum[ EulerPhi[n/k]*3^k*Binomial[ 2k, k], {k, Most[ Divisors[n]]}]) + q[n]; q[n_?OddQ] := 2*(3^((n-1)/2)/(n+1))*Binomial[ n-1, (n-1)/2]; q[n_?EvenQ] := 2*(n-1)*(3^((n-2)/2)/(n*(n+2)))*Binomial[ n-2, (n-2)/2]; Table[ a[n], {n, 0, 21}] (* Jean-Fran├žois Alcover, after Valery A. Liskovets *)


Cf. A000168.

Sequence in context: A030809 A030958 A030885 * A204198 A003322 A170939

Adjacent sequences:  A006381 A006382 A006383 * A006385 A006386 A006387




N. J. A. Sloane


More terms from Alois P. Heinz, Apr 24 2009



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