%I M1281 #48 Dec 19 2021 00:09:40

%S 1,2,4,14,57,312,2071,15030,117735,967850,8268816,72833730,658049140,

%T 6074058060,57106433817,545532037612,5284835906037,51833908183164,

%U 514019531037910,5147924676612282,52017438279806634,529867070532745464

%N Number of planar maps with n edges.

%D 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.

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

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

%D T. R. S. Walsh, personal communication.

%H Alois P. Heinz, <a href="/A006384/b006384.txt">Table of n, a(n) for n = 0..300</a>

%H V. A. Liskovets, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i1r88">Enumerative formulas for unrooted planar maps: a pattern</a>, Electron. J. Combin., 11:1 (2004), R88.

%H Valery A. Liskovets, <a href="http://dx.doi.org/10.1016/0012-365X(94)00347-L">A reductive technique for enumerating non-isomorphic planar maps</a>, Discrete Math. 156 (1996), no. 1-3, 197--217. MR1405018 (97f:05087) - From _N. J. A. Sloane_, Jun 03 2012

%H Timothy R. Walsh, <a href="http://dx.doi.org/10.1137/0604018">Generating nonisomorphic maps without storing them</a>, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.

%H N. J. A. Sloane, <a href="/A006384/a006384.pdf">Notes</a>

%H T. R. S. Walsh, <a href="/A007401/a007401.pdf">Number of sensed planar maps with n edges and m vertices</a>

%H T. R. S. Walsh, <a href="/A007401/a007401_2.pdf">Data (Preprint 1, Part 1)</a>

%H T. R. S. Walsh, <a href="/A007401/a007401_3.pdf">Data (Preprint 1, Part 2)</a>

%H T. R. S. Walsh, <a href="/A007401/a007401_4.pdf">Data (Preprint 1, Part 3)</a>

%H T. R. S. Walsh, <a href="/A007401/a007401_1.pdf">Notes</a>

%H T. R. S. Walsh, <a href="/A007401/a007401.pdf">Number of sensed planar maps with n edges and m vertices</a>

%H T. R. S. Walsh & N. J. A. Sloane, <a href="/A007401/a007401_5.pdf">Correspondence, 1991</a>

%H Nicholas C. Wormald, <a href="http://dx.doi.org/10.1016/0012-365X(81)90238-7">Counting unrooted planar maps</a>, Discrete Math. 36 (1981), no. 2, 205-225.

%H N. C. Wormald, <a href="/A007401/a007401_6.pdf">On the number of planar maps</a>, Can. J. Math. 33.1 (1981), 1-11. (Annotated scanned copy)

%F 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

%F 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

%F a(n) ~ 12^n / (sqrt(Pi) * n^(7/2)). - _Vaclav Kotesovec_, Sep 12 2014

%p 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

%t 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_ *)

%Y Cf. A000168.

%K nonn,nice

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Alois P. Heinz_, Apr 24 2009