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A342053 Array read by antidiagonals: T(n,k) is the number of unrooted 3-connected triangulations of a disk with n interior nodes and k nodes on the boundary, n >= 1, k >= 3.

%I A342053 #9 Mar 13 2021 20:33:54
%S A342053 1,1,1,1,2,4,1,2,8,16,1,3,12,38,78,1,3,20,73,219,457,1,4,27,140,503,
%T A342053 1404,2938,1,4,39,235,1089,3661,9714,20118,1,5,51,392,2149,8796,27715,
%U A342053 70454,144113,1,5,68,610,4050,19419,72204,214664,527235,1065328
%N A342053 Array read by antidiagonals: T(n,k) is the number of unrooted 3-connected triangulations of a disk with n interior nodes and k nodes on the boundary, n >= 1, k >= 3.
%C A342053 For k >= 4, T(n,k) is the number of polyhedra with n+k vertices whose faces are all triangular, except one which is k-gonal.
%C A342053 The initial terms of this sequence can also be computed using the tool "plantri", in particular the command "./plantri -u -v -P [n]" will compute values for a diagonal.
%H A342053 Andrew Howroyd, <a href="/A342053/b342053.txt">Table of n, a(n) for n = 1..1275</a>
%H A342053 G. Brinkmann and B. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/plantri/">Plantri (program for generation of certain types of planar graph)</a>
%H A342053 William G. Brown, <a href="http://dx.doi.org/10.1112/plms/s3-14.4.746">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
%H A342053 Andrew Howroyd, <a href="/A342053/a342053.txt">PARI Program</a>
%e A342053 Array begins:
%e A342053 ===================================================
%e A342053 n\k |     3     4      5      6       7       8
%e A342053 ----+----------------------------------------------
%e A342053   1 |     1     1      1      1       1       1 ...
%e A342053   2 |     1     2      2      3       3       4 ...
%e A342053   3 |     4     8     12     20      27      39 ...
%e A342053   4 |    16    38     73    140     235     392 ...
%e A342053   5 |    78   219    503   1089    2149    4050 ...
%e A342053   6 |   457  1404   3661   8796   19419   40485 ....
%e A342053   7 |  2938  9714  27715  72204  173779  393123 ...
%e A342053   8 | 20118 70454 214664 596906 1538221 3723976 ...
%e A342053   ...
%o A342053 (PARI) A342053Array(8,6) \\ See links for program.
%Y A342053 Columns k=3..6 are A002713, A058786(n+4), A342054, A342055.
%Y A342053 Antidiagonal sums are A342056.
%Y A342053 Cf. A169808 (2-connected), A341856 (rooted), A341923 (oriented).
%K A342053 nonn,tabl
%O A342053 1,5
%A A342053 _Andrew Howroyd_, Feb 26 2021