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A006384 Number of planar maps with n edges.
(Formerly M1281)
5
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)
OFFSET
0,2
REFERENCES
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.
LINKS
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, Notes
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)
FORMULA
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
MAPLE
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
MATHEMATICA
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-François Alcover, after Valery A. Liskovets *)
CROSSREFS
Cf. A000168.
Sequence in context: A030809 A030958 A030885 * A204198 A003322 A170939
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Alois P. Heinz, Apr 24 2009
STATUS
approved

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