A378103 Triangle read by rows: T(n,k) is the number of n-node planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 0, 2, 4, 4, 5, 4, 0, 0, 2, 6, 10, 14, 14, 18, 16, 0, 0, 0, 7, 18, 35, 49, 63, 69, 88, 78

Offset

3,8

Comments

The planar maps considered are connected, unsensed and with no loops or isthmuses. (A loop corresponds to an internal face of size 1 and an isthmus corresponds to an internal face of size 2).

In other words, a(n) is the number of embeddings in the plane of connected bridgeless planar simple graphs with n vertices and k triangular internal faces.

The number of edges is n + k - 1.

The nonzero terms in row n range from k = floor(n/2) through 2*n-5 and, thus, the number of nonzero terms is 2n - floor(n/2) - 4 = A001651(n-2).

Links

Table of n, a(n) for n=3..38.

Ya-Ping Lu, Illustration of initial terms

Formula

T(n, 2*n-5) = A002713(n-3).

Example

Triangle begins:
n\k        1     2     3     4     5     6     7     8     9    10    11
----     ----  ----  ----  ----  ----  ----  ----  ----  ----  ----  ----
3          1
4          0     1     1
5          0     1     1     2     1
6          0     0     2     4     4     5     4
7          0     0     2     6    10    14    14    18    16
8          0     0     0     7    18    35    49    63    69    88    78

Crossrefs

Row sums are A377785.

Cf. A001651, A002713, A003094, A169808.

Sequence in context: A059220 A261630 A301503 * A059431 A289358 A271698

Adjacent sequences: A378098 A378101 A378102 * A378104 A378106 A378107

Keyword

nonn,tabf,more,new

Author

Ya-Ping Lu, Nov 16 2024

Status

approved