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A341856 Array read by antidiagonals: T(n,k) is the number of rooted strong triangulations of a disk with n interior nodes and 3+k nodes on the boundary. 9
1, 0, 1, 0, 1, 3, 0, 1, 6, 13, 0, 1, 10, 36, 68, 0, 1, 15, 80, 228, 399, 0, 1, 21, 155, 610, 1518, 2530, 0, 1, 28, 273, 1410, 4625, 10530, 16965, 0, 1, 36, 448, 2933, 12165, 35322, 75516, 118668, 0, 1, 45, 696, 5628, 28707, 102548, 272800, 556512, 857956 (list; table; graph; refs; listen; history; text; internal format)
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
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
Columns k=0..3 are A000260, A242136, A341917, A341918.
Antidiagonal sums give A341919.
Cf. A146305 (not necessarily strong triangulations), A210664, A341923, A342053.
Sequence in context: A243984 A100485 A143397 * A339350 A244118 A273155
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Feb 23 2021
STATUS
approved

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Last modified May 28 01:34 EDT 2024. Contains 372900 sequences. (Running on oeis4.)