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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

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

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

Adjacent sequences:  A341853 A341854 A341855 * A341857 A341858 A341859

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Feb 23 2021

STATUS

approved

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Last modified June 29 17:30 EDT 2022. Contains 354913 sequences. (Running on oeis4.)