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%I #10 Mar 14 2021 18:48:15
%S 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,
%T 1518,2530,0,1,28,273,1410,4625,10530,16965,0,1,36,448,2933,12165,
%U 35322,75516,118668,0,1,45,696,5628,28707,102548,272800,556512,857956
%N 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.
%C 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.
%H Andrew Howroyd, <a href="/A341856/b341856.txt">Table of n, a(n) for n = 0..1325</a>
%H William T. Tutte, <a href="http://dx.doi.org/10.4153/CJM-1962-002-9">A census of planar triangulations</a>, Canad. J. Math. 14 (1962), 21-38.
%F T(n,0) = A000260(n) = 2*(4*n+1)!/((3*n+2)!*(n+1)!).
%F 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.
%e Array begins:
%e =======================================================
%e n\k | 0 1 2 3 4 5 6
%e ----+--------------------------------------------------
%e 0 | 1 0 0 0 0 0 0 ...
%e 1 | 1 1 1 1 1 1 1 ...
%e 2 | 3 6 10 15 21 28 36 ...
%e 3 | 13 36 80 155 273 448 696 ...
%e 4 | 68 228 610 1410 2933 5628 10128 ...
%e 5 | 399 1518 4625 12165 28707 62230 125928 ...
%e 6 | 2530 10530 35322 102548 267162 638624 1422204 ...
%e ...
%o (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)!)))
%Y Columns k=0..3 are A000260, A242136, A341917, A341918.
%Y Antidiagonal sums give A341919.
%Y Cf. A146305 (not necessarily strong triangulations), A210664, A341923, A342053.
%K nonn,tabl
%O 0,6
%A _Andrew Howroyd_, Feb 23 2021
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