cp's OEIS Frontend

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

%I A380237 #10 Jan 21 2025 21:29:50
%S A380237 1,2,5,14,42,140,473,1670,5969,21679,79419,293496,1091006,4078213,
%T A380237 15312150,57721030,218333832,828408842,3151769615,12020870753,
%U A380237 45949957412,176001205559,675384194565,2596119292840,9994894356158,38535398284100,148772774499015,575079507042663
%N A380237 Number of sensed planar maps with n vertices and 2 faces.
%C A380237 Also, by duality the number of sensed planar maps with n faces and 2 vertices.
%C A380237 The number of edges is n.
%H A380237 Andrew Howroyd, <a href="/A380237/b380237.txt">Table of n, a(n) for n = 1..1000</a>
%F A380237 a(n) = (A210736(n) + A060404(n))/2.
%F A380237 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))).
%F A380237 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.
%o A380237 (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}
%o A380237 (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}
%Y A380237 Column 2 of A379430.
%Y A380237 Cf. A000346 (rooted), A380238 (achiral), A380239 (unsensed), A060404 (with a distinguished face), A103943 (with a distinguished vertex).
%Y A380237 Cf. A000108, A210736.
%K A380237 nonn
%O A380237 1,2
%A A380237 _Andrew Howroyd_, Jan 19 2025