A006390 Number of unrooted loopless planar n-edge maps. (Formerly M1468)

1, 1, 2, 5, 14, 49, 240, 1259, 7570, 47996, 319518, 2199295, 15571610, 112773478, 832809504, 6253673323, 47650870538, 367784975116, 2871331929096, 22647192990256, 180277915464664, 1447060793168493, 11703567787559680, 95312765368320637, 781151020141584190

Offset

0,3

Comments

By duality, also the number of unrooted (sensed) isthmusless planar n-edge maps. An isthmus may also be called a bridge. - Andrew Howroyd, Mar 28 2021

References

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

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

V. A. Liskovets and T. R. S. Walsh, Counting Unrooted Loopless Planar Maps [Extended abstract]

V. A. Liskovets and T. R. S. Walsh, Counting unrooted loopless planar maps, Europ. J. Combin., 26:5 (2005), 651-663.

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

Formula

a(n) = (1/(2n))*[2(4n+1)*binomial(4n, n)/((n+1)*(3n+1)*(3n+2)) + Sum_{0<k<n, k|n}phi(n/k)*binomial(4k, k)+q(n)] where phi is the Euler function (A000010), q(n)=binomial(2n, (n-2)/2) if n is even and q(n)=2n*binomial(2n, (n-1)/2)/(n+1) if n is odd.

Mathematica

a[n_] := If[n==0, 1, (1/(2n))(Sum[Binomial[4k, k] EulerPhi[n/k] Boole[ 0<k<n], {k, Divisors[n]}] + 2(4n+1) Binomial[4n, n]/((n+1)(3n+1)(3n+2)) + q[n])];
q[n_] := If[EvenQ[n], Binomial[2n, (n-2)/2], 2n Binomial[2n, (n-1)/2]/ (n+1)];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 29 2019 *)

Prog

(PARI) a(n) = {if(n==0, 1, (sumdiv(n, d, if(d<n, 1, 2*(4*n+1)/((n+1)*(3*n+1)*(3*n+2))) * eulerphi(n/d) * binomial(4*d, d)) + if(n%2, 2*n/(n+1), 1)*binomial(2*n, (n-1)\2))/(2*n))} \\ Andrew Howroyd, Mar 28 2021

Crossrefs

Cf. A000010, A000260 (rooted), A006391 (unsensed case), A103941 (with distinguished face), A103942 (with distinguished vertex).

Sequence in context: A079452 A081920 A224978 * A100597 A352184 A022562

Adjacent sequences: A006387 A006388 A006389 * A006391 A006392 A006393

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Valery A. Liskovets, Dec 01 2003

a(17) and a(19) corrected by Sean A. Irvine, Mar 26 2017

Status

approved