cp's OEIS Frontend

A006390 Number of sensed loopless planar maps with n edges.

%I A006390 M1468 #38 Jan 16 2025 11:30:44
%S A006390 1,1,2,5,14,49,240,1259,7570,47996,319518,2199295,15571610,112773478,
%T A006390 832809504,6253673323,47650870538,367784975116,2871331929096,
%U A006390 22647192990256,180277915464664,1447060793168493,11703567787559680,95312765368320637,781151020141584190
%N A006390 Number of sensed loopless planar maps with n edges.
%C A006390 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
%D A006390 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006390 Andrew Howroyd, <a href="/A006390/b006390.txt">Table of n, a(n) for n = 0..500</a>
%H A006390 V. A. Liskovets and T. R. S. Walsh, <a href="https://web.archive.org/web/20051104034322/http://www.pims.math.ca:80/science/2004/fpsac/papers/Liskovets.pdf">Counting Unrooted Loopless Planar Maps</a> [Extended abstract]
%H A006390 V. A. Liskovets and T. R. S. Walsh, <a href="https://doi.org/10.1016/j.ejc.2004.04.011">Counting unrooted loopless planar maps</a>, Europ. J. Combin., 26:5 (2005), 651-663.
%H A006390 Timothy R. Walsh, <a href="https://doi.org/10.1137/0604018">Generating nonisomorphic maps without storing them</a>, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
%F A006390 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.
%t A006390 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])];
%t A006390 q[n_] := If[EvenQ[n], Binomial[2n, (n-2)/2], 2n Binomial[2n, (n-1)/2]/ (n+1)];
%t A006390 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Aug 29 2019 *)
%o A006390 (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
%Y A006390 Cf. A000010, A006384, A000260 (rooted), A006391 (unsensed case), A103941 (with distinguished face), A103942 (with distinguished vertex).
%K A006390 nonn
%O A006390 0,3
%A A006390 _N. J. A. Sloane_
%E A006390 More terms from _Valery A. Liskovets_, Dec 01 2003
%E A006390 a(17) and a(19) corrected by _Sean A. Irvine_, Mar 26 2017