A378340 Triangle read by rows: T(n,k) is the number of n node connected achiral 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, 3, 3, 4, 3, 0, 0, 2, 4, 7, 8, 7, 10, 8, 0, 0, 0, 4, 8, 15, 19, 22, 19, 29, 23, 0, 0, 0, 3, 11, 22, 32, 48, 57, 65, 57, 86, 68, 0, 0, 0, 0, 8, 25, 47, 82, 104, 150, 175, 200, 176, 266, 215, 0, 0, 0, 0, 7, 26, 64, 123, 186, 288, 346, 488, 556, 634, 557, 844, 680

Offset: 3

Author

Andrew Howroyd, Nov 25 2024

Keywords

• nonn
• tabf

Comments

See A378103 for illustration of initial terms. This sequence counts only those maps which have mirror symmetry.

The planar maps considered are without loops or isthmuses.

The number of edges is n + k - 1.

Examples

			Triangle begins:
n\k | 1  2  3  4   5   6   7   8   9  10  11  12  13
----+------------------------------------------------
  3 | 1;
  4 | 0, 1, 1;
  5 | 0, 1, 1, 2,  1;
  6 | 0, 0, 2, 3,  3,  4,  3;
  7 | 0, 0, 2, 4,  7,  8,  7, 10,  8;
  8 | 0, 0, 0, 4,  8, 15, 19, 22, 19, 29, 23;
  9 | 0, 0, 0, 3, 11, 22, 32, 48, 57, 65, 57, 86, 68;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 3..2306 (rows 3..50)
• Andrew Howroyd, PARI Program, Nov 2024.

Crossrefs

Row sums are A378339.

Column sums are A378341.

Antidiagonal sums are A378342.

Cf. A378103 (unsensed), A378336 (sensed), A002712.

Programs

PARI

my(A=A378340rows(10)); for(i=1, #A, print(A[i])) \\ See Links for program.

Formula

T(n,2*n-5) = A002712(n-3). - Ya-Ping Lu, Dec 16 2024