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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

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.

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

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-Fran├žois Alcover, after Valery A. Liskovets *)

CROSSREFS

Cf. A000168.

Sequence in context: A030809 A030958 A030885 * A204198 A003322 A170939

Adjacent sequences:  A006381 A006382 A006383 * A006385 A006386 A006387

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Alois P. Heinz, Apr 24 2009

STATUS

approved

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Last modified November 22 15:27 EST 2017. Contains 295089 sequences.