OFFSET
0,6
COMMENTS
A strong triangulation is one in which no interior edge joins two nodes on the boundary. Except for the single triangle which is enumerated by T(0,0) these are the 3-connected triangulations.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38.
FORMULA
T(n,0) = A000260(n) = 2*(4*n+1)!/((3*n+2)!*(n+1)!).
T(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)!) for m > 0.
EXAMPLE
Array begins:
=======================================================
n\k | 0 1 2 3 4 5 6
----+--------------------------------------------------
0 | 1 0 0 0 0 0 0 ...
1 | 1 1 1 1 1 1 1 ...
2 | 3 6 10 15 21 28 36 ...
3 | 13 36 80 155 273 448 696 ...
4 | 68 228 610 1410 2933 5628 10128 ...
5 | 399 1518 4625 12165 28707 62230 125928 ...
6 | 2530 10530 35322 102548 267162 638624 1422204 ...
...
PROG
(PARI) T(n, m)=if(m==0, 2*(4*n+1)!/((3*n+2)!*(n+1)!), (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)!)))
CROSSREFS
Antidiagonal sums give A341919.
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Feb 23 2021
STATUS
approved