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

0,4

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)

Links

Table of n, a(n) for n=0..65.

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, ...

...

Example

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.

Sequence in context: A033185 A217781 A339428 * A105632 A091491 A117418

Adjacent sequences: A204846 A204847 A204848 * A204850 A204851 A204852

Keyword

nonn,tabl

Author

Gary W. Adamson, Jan 19 2012

Status

approved