A103941 -id:A103941 - cp's OEIS frontend

A006390 Number of sensed loopless planar maps with n edges.

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

N. J. A. Sloane

nonn

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

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

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.

Cf. A000010, A006384, A000260 (rooted), A006391 (unsensed case), A103941 (with distinguished face), A103942 (with distinguished vertex).

a[n_] := If[n==0, 1, (1/(2n))(Sum[Binomial[4k, k] EulerPhi[n/k] Boole[ 0Jean-François Alcover, Aug 29 2019 *)

a(n) = {if(n==0, 1, (sumdiv(n, d, if(dAndrew Howroyd, Mar 28 2021

a(n) = (1/(2n))*[2(4n+1)*binomial(4n, n)/((n+1)*(3n+1)*(3n+2)) + Sum_{0A000010), 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.

More terms from Valery A. Liskovets, Dec 01 2003

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

A103942 Number of unrooted n-edge isthmusless maps in the plane (planar with a distinguished outside face).

Original entry on oeis.org

1, 1, 3, 9, 38, 187, 1120, 7083, 47990, 337676, 2455517, 18310155, 139447034, 1080773098, 8502896424, 67763884363, 546147639926, 4445389286380, 36501274080076, 302060508150976, 2517213486505592, 21110062391001119, 178052027949519768, 1509631210682469661, 12860805940582898474

Offset: 0

Valery A. Liskovets, Mar 17 2005

V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Andrew Howroyd, Table of n, a(n) for n = 0..500
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Cf. A027836, A006390, A103941, A000260.

a[n_] := (1/(2n)) ((5n^2 + 13n + 2) Binomial[4n, n]/((n+1)(3n+1)(3n+2)) + Sum[Boole[0 < k < n] EulerPhi[n/k] Binomial[4k, k], {k, Divisors[n]}] + q[n]);
q[n_] := If[EvenQ[n], 0, (n-1) Binomial[2n, (n-1)/2]]/(n+1);
Array[a, 20] (* Jean-François Alcover, Sep 01 2019 *)

a(n) = {if(n==0, 1, (sumdiv(n, d, if(dAndrew Howroyd, Mar 28 2021

For n > 0, a(n) = (1/(2n))*[(5n^2+13n+2)*binomial(4n, n)/((n+1)(3n+1)(3n+2)) + Sum_{0A000010), q(n)=0 if n is even and q(n)=(n-1)*binomial(2n, (n-1)/2)/(n+1) if n is odd.

a(0)=1 prepended and terms a(21) and beyond from Andrew Howroyd, Mar 28 2021

cp's OEIS Frontend

A006390 Number of sensed loopless planar maps with n edges.

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