A380237 Number of sensed planar maps with n vertices and 2 faces.

1, 2, 5, 14, 42, 140, 473, 1670, 5969, 21679, 79419, 293496, 1091006, 4078213, 15312150, 57721030, 218333832, 828408842, 3151769615, 12020870753, 45949957412, 176001205559, 675384194565, 2596119292840, 9994894356158, 38535398284100, 148772774499015, 575079507042663

Offset: 1

Author

Andrew Howroyd, Jan 19 2025

Keywords

• nonn

Comments

Also, by duality the number of sensed planar maps with n faces and 2 vertices.

The number of edges is n.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

Crossrefs

Column 2 of A379430.

Cf. A000346 (rooted), A380238 (achiral), A380239 (unsensed), A060404 (with a distinguished face), A103943 (with a distinguished vertex).

Cf. A000108, A210736.

Programs

PARI

a(n) = {(binomial(n - 1, (n - 1)\2) + sumdiv(n, d, eulerphi(n/d)*(2^(2*d-1) - binomial(2*d-1, d)))/n)/2}

PARI

seq(n)={my(c(d)=(1-sqrt(1-4*x^d + O(x*x^(n+d))))/(2*x^d)); Vec(1/(1 - x*c(2)) - 1 - sum(k=1, n, log(2 - c(k))*eulerphi(k)/k))/2}

Formula

a(n) = (A210736(n) + A060404(n))/2.

a(n) = (1/(2*n))*(n*binomial(n-1, floor((n-1)/2)) + Sum_{d|n} phi(n/d)*(2^(2*d-1) - binomial(2*d-1, d))).

G.f.: (1/2)*(1/(1 - x*C(x^2)) - 1 - Sum_{k>=1} log(1 - C(x^k)) * phi(k)/k), where C(x) is the g.f. of A000108.