A210664 Square array read by upwards antidiagonals: T(m,n) is the number of simple 3-connected triangulations of a closed region in the plane with m+3 given external edges and 3n+m internal edges, m>=0, n>=1.

1, 1, 0, 1, 2, 1, 1, 5, 6, 3, 1, 9, 20, 22, 12, 1, 14, 50, 85, 91, 52, 1, 20, 105, 254, 385, 408, 241, 1, 27, 196, 644, 1287, 1836, 1938, 1173, 1, 35, 336, 1448, 3696, 6630, 9120, 9614, 5929, 1, 44, 540, 2967, 9468, 20790, 34846, 46805, 49335, 30880

Offset

0,5

Comments

A triangulation is simple if it contains no separating 3-cycle. There are n interior nodes and m+3 nodes on the boundary. - Andrew Howroyd, Feb 24 2021

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325

P. N. Rathie, A census of simple planar triangulations, J. Combin. Theory, B 16 (1974), 134-138.

William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38.

Formula

From Andrew Howroyd, Feb 24 2021: (Start)

G.f. of row m > 0: R(x) satisfies g(x^2)^(m+1)*R(x*g(x^2)) = B(x^2) where g(x) is the g.f. of column 0 of A341856 and B(x) is the g.f. of column m of A341856.

G.f. of row m > 0: h(x)^(m+1)*B(x*h(x)^2) where 2-h(x) is the g.f. of A000256 and B(x) is the g.f. of column m of A341856.

(End)

Example

Array begins:
  1,  0,   1,    3,   12, ... (A000256)
  1,  2,   6,   22,   91, ...
  1,  5,  20,   85,  385, ...
  1,  9,  50,  254, 1287, ...
  1, 14, 105,  644, 3696, ...
  1, 20, 196, 1448, 9468, ...
  ...
From Andrew Howroyd, Feb 24 2021: (Start)
The array transposed for comparability with A341856 begins:
==================================================
n\m |   0    1    2     3      4      5      6
----+---------------------------------------------
  1 |   1    1    1     1      1      1      1 ...
  2 |   0    2    5     9     14     20     27 ...
  3 |   1    6   20    50    105    196    336 ...
  4 |   3   22   85   254    644   1448   2967 ...
  5 |  12   91  385  1287   3696   9468  22131 ...
  6 |  52  408 1836  6630  20790  58564 151146 ...
  7 | 241 1938 9120 34846 116641 353056 983664 ...
(End)

Prog

(PARI) \\ here H is A000256 as g.f., U(n, m) is A341856 for m > 0.
H(n)={my(g=1+serreverse(x/(1+x)^4 + O(x*x^n) )); 2 - sqrt(serreverse(x*(2-g)^2*g^4)/x)}
U(n, m)={(3*(m+2)!*(m-1)!/(3*n+3*m+3)!)*sum(j=0, min(m, n-1), (4*n+3*m-j+1)!*(m+j+2)*(m-3*j)/(j!*(j+1)!*(m-j)!*(m-j+2)!*(n-j-1)!))}
R(N, m)={my(g=2-H(N)); Vec(if(m==0, 1-g, g^(m+1)*subst(O(x*x^N) + sum(n=1, N, U(n, m)*x^n), x, x*g^2)))}
M(m, n=m)={Mat(vectorv(m+1, i, R(n, i-1)))}
M(7) \\ Andrew Howroyd, Feb 23 2021

Crossrefs

Rows m=0..3 are A000256, A000139, A341920, A341921.

Columns are A000012, A000096, A002415, A004305.

Antidiagonal sums give A341922.

Cf. A341856.

Sequence in context: A293219 A266572 A266681 * A176093 A092437 A064814

Adjacent sequences: A210661 A210662 A210663 * A210665 A210666 A210667

Keyword

nonn,tabl

Author

N. J. A. Sloane, Mar 28 2012

Extensions

Terms a(21) and beyond from Andrew Howroyd, Feb 23 2021

Status

approved