A204849 A Motzkin triangle by rows.

1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 6, 4, 1, 1, 21, 15, 8, 5, 1, 1, 51, 36, 22, 10, 6, 1, 1, 127, 91, 54, 30, 12, 7, 1, 1, 323, 232, 142, 75, 39, 14, 8, 1, 1, 835, 603, 370, 205, 99, 49, 16, 9, 1, 1, 2188, 1585, 983, 545, 281, 126, 60, 18, 10, 1, 1

Offset: 0

Author

Gary W. Adamson, Jan 19 2012

Keywords

• nonn
• tabl

Comments

Left border = A001006, row sums = A001006 with offset 1.

From R. J. Mathar, Jul 24 2017: (Start)

The element T(n-1,k) counts the RGS's in Arndt's bijection of Apr 17 2013 in A001006 which have length n and finish with the k-th largest possible rise in the last step (0, 2, 3, 4, 5, ..., 1 impossible).

Example with n=4: the four RGS's 0000, 0022, 0033 and 0222 finish with a rise of 0 [T(3,0)=4]; the three RGS's 0002, 0024, 0224 finish with a rise of 2 [T(3,1)=3]; the one RGS 0003 finishes with a rise of 3 [T(3,2)=1]; the one 0004 finishes with a rise of 4 [T(3,3)=1]. (End)

Examples

			First few rows of the triangle =
     1;
     1,    1;
     2,    1,   1;
     4,    3,   1,   1;
     9,    6,   4,   1,   1;
    21,   15,   8,   5,   1,   1;
    51,   36,  22,  10,   6,   1,  1;
   127,   91,  54,  30,  12,   7,  1,  1;
   323,  232, 142,  75,  39,  14,  8,  1,  1;
   835,  603, 370, 205,  99,  49, 16,  9,  1,  1;
  2188, 1585, 983, 545, 281, 126, 60, 18, 10,  1, 1;
  ...
Top row of M^3 = [4, 3, 1, 1, 0, 0, 0, ...].
		

Crossrefs

Cf. A001006.

Formula

n-th row of the triangle is the top row of M^n (deleting the zeros), where M = the following infinite square production matrix:

1, 1, 0, 0, 0, 0, 0, ...

1, 0, 1, 0, 0, 0, 0, ...

1, 1, 0, 1, 0, 0, 0, ...

1, 1, 1, 0, 1, 0, 0, ...

1, 1, 1, 1, 0, 1, 0, ...

1, 1, 1, 1, 1, 0, 1, ...

...