A378103 Triangle read by rows: T(n,k) is the number of n-node connected unsensed 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, 0, 0, 0, 5, 28, 74, 131, 204, 274, 345, 396, 489, 457, 0, 0, 0, 0, 26, 126, 304, 574, 893, 1290, 1708, 2137, 2503, 3071, 2938, 0, 0, 0, 0, 13, 159, 582, 1396, 2613, 4274, 6270, 8709, 11433, 14227, 16905, 20667, 20118

Offset: 3

Author

Ya-Ping Lu, Nov 16 2024

Keywords

• nonn
• tabf

Comments

The planar maps considered are without loops or isthmuses.

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).

Examples

			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
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 3..2306 (rows 3..50)
• Ya-Ping Lu, Illustration of initial terms

Crossrefs

Row sums are A377785.

Cf. A001651, A002713, A003094, A169808, A378336 (sensed), A378340 (achiral).

The final 3 terms of each row are in A002713, A005500, A005501.

Programs

PARI

my(A=A378103rows(10)); for(i=1, #A, print(A[i])) \\ See PARI link in A378340 for program code. - Andrew Howroyd, Nov 25 2024

Formula

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

T(n,k) = (A378336(n,k) + A378340(n,k))/2.

Extensions

a(39) onwards from Andrew Howroyd, Nov 25 2024