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.

1, 1, 1, 1, 2, 1, 4, 5, 6, 5, 6, 16, 21, 26, 24, 19, 48, 88, 119, 147, 133, 49, 164, 330, 538, 735, 892, 846, 150, 559, 1302, 2310, 3568, 4830, 5876, 5661, 442, 1952, 5005, 9882, 16500, 24596, 33253, 40490, 39556, 1424, 6872, 19504, 41715, 75387, 120582, 176354, 237336, 290020, 286000, 4522

Offset

0,5

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).

Jean-François Alcover, Mathematica code

W. G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]. See Table 1 (with a typo at G(n=1,m=6)).

L. March and C. F. Earl, On Counting Architectural Plans, Environment and Planning B, 4 (1977), 57-80. See Table 2.

Formula

Brown (Eq. 6.3) gives a formula.

Example

Array begins:
 ==============================================================
 n\k |    0     1      2       3       4        5         6 ...
 ----+---------------------------------------------------------
   0 |    1     1      1       4       6       19        49 ...
   1 |    1     2      5      16      48      164       559 ...
   2 |    1     6     21      88     330     1302      5005 ...
   3 |    5    26    119     538    2310     9882     41715 ...
   4 |   24   147    735    3568   16500    75387    338685 ...
   5 |  133   892   4830   24596  120582   578622   2730728 ...
   6 |  846  5876  33253  176354  900240  4493168  22037055 ...
   7 | 5661 40490 237336 1298732 6849810 35286534 178606610 ...
   ...
The first few antidiagonals are:
  1,
  1,1,
  1,2,1,
  4,5,6,5,
  6,16,21,26,24,
  19,48,88,119,147,133,
  49,164,330,538,735,892,846,
  ...

Maple

A262586 := proc(n, m)
    BrownG(n, m) ; # procedure in A210696
end proc:
for d from 0 to 12 do
    for n from 0 to d do
        printf("%d, ", A262586(n, d-n)) ;
    end do:
end do: # R. J. Mathar, Oct 21 2015

Mathematica

See LINKS section.

Prog

(PARI) \\ See Links in A169808 for PARI program file.
{ for(n=0, 7, for(k=0, 7, print1(OrientedTriangs(n, k), ", ")); print) } \\ Andrew Howroyd, Nov 23 2024

Crossrefs

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

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

Antidiagonal sums are A341855.

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

Sequence in context: A074720 A323456 A326058 * A058359 A261608 A351250

Adjacent sequences: A262583 A262584 A262585 * A262587 A262588 A262589

Keyword

nonn,tabl,changed

Author

N. J. A. Sloane, Oct 20 2015

Status

approved