A262586 - OEIS

%I #33 Nov 23 2024 14:27:34

%S 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 538,735,892,846,150,559,1302,2310,3568,4830,5876,5661,442,1952,5005,

%U 9882,16500,24596,33253,40490,39556,1424,6872,19504,41715,75387,120582,176354,237336,290020,286000,4522

%N 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 Andrew Howroyd, <a href="/A262586/b262586.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).

%H Jean-François Alcover, <a href="/A262586/a262586.txt">Mathematica code</a>

%H 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 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 Brown (Eq. 6.3) gives a formula.

%e Array begins:

%e  ==============================================================

%e  n\k |    0     1      2       3       4        5         6 ...

%e  ----+---------------------------------------------------------

%e    0 |    1     1      1       4       6       19        49 ...

%e    1 |    1     2      5      16      48      164       559 ...

%e    2 |    1     6     21      88     330     1302      5005 ...

%e    3 |    5    26    119     538    2310     9882     41715 ...

%e    4 |   24   147    735    3568   16500    75387    338685 ...

%e    5 |  133   892   4830   24596  120582   578622   2730728 ...

%e    6 |  846  5876  33253  176354  900240  4493168  22037055 ...

%e    7 | 5661 40490 237336 1298732 6849810 35286534 178606610 ...

%e    ...

%e The first few antidiagonals are:

%e   1,

%e   1,1,

%e   1,2,1,

%e   4,5,6,5,

%e   6,16,21,26,24,

%e   19,48,88,119,147,133,

%e   49,164,330,538,735,892,846,

%e   ...

%p A262586 := proc(n,m)

%p     BrownG(n,m) ; # procedure in A210696

%p end proc:

%p for d from 0 to 12 do

%p     for n from 0 to d do

%p         printf("%d,",A262586(n,d-n)) ;

%p     end do:

%p end do: # _R. J. Mathar_, Oct 21 2015

%t See LINKS section.

%o (PARI) \\ See Links in A169808 for PARI program file.

%o { for(n=0, 7, for(k=0, 7, print1(OrientedTriangs(n,k), ", ")); print) } \\ _Andrew Howroyd_, Nov 23 2024

%Y Columns 0..2 are A002709, A002710, A002711.

%Y Rows 0..3 are A001683, A210696, A005498, A005499.

%Y Antidiagonal sums are A341855.

%Y Cf. A169808 (unoriented), A169809 (achiral).

%K nonn,tabl

%O 0,5

%A _N. J. A. Sloane_, Oct 20 2015