A384967 Number of unsensed simple planar maps with n vertices and 2 faces.

0, 0, 1, 2, 7, 22, 76, 271, 1001, 3765, 14381, 55450, 214880, 835663, 3255652, 12698352, 49559793, 193513944, 755852101, 2953214386, 11541989533, 45123241746, 176465152051, 690340349398, 2701579878022, 10576116931462, 41418132927403, 162259989848094, 635899817853002, 2492993368347594

Offset

1,4

Comments

In other words, a(n) is the number of embeddings on the sphere of connected simple unicyclic planar graphs with n nodes.

Links

Table of n, a(n) for n=1..30.

Prog

(PARI)
G1(n)={my(g=(1-sqrt(1-4*x^2 + O(x^(n+2))))/(2*x^2)); ((1 + x/(1-x-x^2*g)^2)^2/(1 - x^2*g^2) - 1)/2 + 1/(1 - x*g) - 1 - x*(g^2/(1 - x*g)^2 + g) - x^2*(g^4/(1 - x*g)^4 + 3*g^2)/2}
G2(n)={my(c(d)=(1-sqrt(1-4*x^d + O(x*x^(n+d))))/(2*x^d)); sum(k=1, n, my(m=1+k%2); -(log(2 - c(k)) + log(1 - x^k*c(m*k)^(2/m)))*eulerphi(k)/k, O(x*x^n)) - x*(c(1)^2 + c(2)) - x^2*(c(1)^4 + 3*c(2)^2)/2}
seq(n)={Vec(G1(n)+G2(n), -n)/4}

Crossrefs

Column 2 of A384963.

Also subdiagonal of A379430.

Cf. A001429, A006081 (cycle is loop), A380239 (not necessarily simple), A384966 (sensed version).

Sequence in context: A387063 A090831 A174403 * A119975 A106188 A176612

Adjacent sequences: A384964 A384965 A384966 * A384968 A384969 A384970

Keyword

nonn

Author

Andrew Howroyd, Jun 15 2025

Status

approved