A384966 -id:A384966 - cp's OEIS frontend

A384964 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of connected simple planar graphs with n nodes and k faces up to orientation preserving isomorphisms, n >= 1, k=1..max(1,2*n-4).

1, 1, 1, 1, 2, 2, 1, 1, 3, 8, 8, 6, 2, 1, 6, 29, 60, 73, 52, 25, 6, 2, 14, 113, 388, 768, 903, 728, 379, 136, 26, 6, 34, 444, 2303, 6584, 11782, 14321, 12113, 7298, 3048, 872, 147, 17, 95, 1763, 12650, 49806, 123547, 210314, 255884, 228807, 150929, 73428, 25536, 6142, 892, 73

Offset: 1

Andrew Howroyd, Jun 13 2025

Equivalently, T(n,k) is the number of sensed simple planar maps with n vertices and k faces.
The number of edges is n+k-2.
Terms of this sequence can be computed using the tool "plantri". The expanded reference gives rows 1..14 of this table.

			Triangle begins:
   1;
   1;
   1,   1,
   2,   2,    1,    1,
   3,   8,    8,    6,     2,     1,
   6,  29,   60,   73,    52,    25,     6,    2,
  14, 113,  388,  768,   903,   728,   379,  136,   26,   6,
  34, 444, 2303, 6584, 11782, 14321, 12113, 7298, 3048, 872, 147, 17;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..158 (rows 1..14)
Gunnar Brinkmann and Brendan McKay, Fast generation of planar graphs (expanded edition), Tables 19-22.

Row sums are A384965.
Antidiagonal sums are A006394.
Columns 1..2 are A002995, A384966.
Cf. A379430 (not necessarily simple), A342059 (2-connected), A239893 (3-connected), A384963 (unsensed).

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

Original entry on oeis.org

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

Andrew Howroyd, Jun 15 2025

nonn

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

Column 2 of A384963.
Also subdiagonal of A379430.
Cf. A001429, A006081 (cycle is loop), A380239 (not necessarily simple), A384966 (sensed version).

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}

cp's OEIS Frontend

A384964 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of connected simple planar graphs with n nodes and k faces up to orientation preserving isomorphisms, n >= 1, k=1..max(1,2*n-4).

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