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A262586 Square array T(n,m) (n>=0, m>=0) read by antidiagonals downwards giving number of rooted triangulations of type [n,m] up to orientation-preserving isomorphisms.

%I A262586 #36 Aug 08 2025 01:54:34
%S A262586 1,1,1,1,2,1,4,5,6,5,6,16,21,26,24,19,48,88,119,147,133,49,164,330,
%T A262586 538,735,892,846,150,559,1302,2310,3568,4830,5876,5661,442,1952,5005,
%U A262586 9882,16500,24596,33253,40490,39556,1424,6872,19504,41715,75387,120582,176354,237336,290020,286000,4522
%N A262586 Square array T(n,m) (n>=0, m>=0) read by antidiagonals downwards giving number of rooted triangulations of type [n,m] up to orientation-preserving isomorphisms.
%H A262586 Andrew Howroyd, <a href="/A262586/b262586.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%H A262586 Jean-François Alcover, <a href="/A262586/a262586.txt">Mathematica code</a>
%H A262586 W. G. Brown, <a href="/A002709/a002709.pdf">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]. See Table 1 (with a typo at G(n=1,m=6)).
%H A262586 L. March and C. F. Earl, <a href="https://doi.org/10.1068/b040057">On Counting Architectural Plans</a>, Environment and Planning B, 4 (1977), 57-80. See Table 2.
%F A262586 Brown (Eq. 6.3) gives a formula.
%e A262586 Array begins:
%e A262586  ==============================================================
%e A262586  n\k |    0     1      2       3       4        5         6 ...
%e A262586  ----+---------------------------------------------------------
%e A262586    0 |    1     1      1       4       6       19        49 ...
%e A262586    1 |    1     2      5      16      48      164       559 ...
%e A262586    2 |    1     6     21      88     330     1302      5005 ...
%e A262586    3 |    5    26    119     538    2310     9882     41715 ...
%e A262586    4 |   24   147    735    3568   16500    75387    338685 ...
%e A262586    5 |  133   892   4830   24596  120582   578622   2730728 ...
%e A262586    6 |  846  5876  33253  176354  900240  4493168  22037055 ...
%e A262586    7 | 5661 40490 237336 1298732 6849810 35286534 178606610 ...
%e A262586    ...
%e A262586 The first few antidiagonals are:
%e A262586   1,
%e A262586   1,1,
%e A262586   1,2,1,
%e A262586   4,5,6,5,
%e A262586   6,16,21,26,24,
%e A262586   19,48,88,119,147,133,
%e A262586   49,164,330,538,735,892,846,
%e A262586   ...
%p A262586 A262586 := proc(n,m)
%p A262586     BrownG(n,m) ; # procedure in A210696
%p A262586 end proc:
%p A262586 for d from 0 to 12 do
%p A262586     for n from 0 to d do
%p A262586         printf("%d,",A262586(n,d-n)) ;
%p A262586     end do:
%p A262586 end do: # _R. J. Mathar_, Oct 21 2015
%t A262586 (* See LINKS section. *)
%o A262586 (PARI) \\ See Links in A169808 for PARI program file.
%o A262586 { for(n=0, 7, for(k=0, 7, print1(OrientedTriangs(n,k), ", ")); print) } \\ _Andrew Howroyd_, Nov 23 2024
%Y A262586 Columns 0..2 are A002709, A002710, A002711.
%Y A262586 Rows 0..3 are A001683, A210696, A005498, A005499.
%Y A262586 Antidiagonal sums are A341855.
%Y A262586 Cf. A169808 (unoriented), A169809 (achiral).
%K A262586 nonn,tabl
%O A262586 0,5
%A A262586 _N. J. A. Sloane_, Oct 20 2015