A218757 Triangle read by rows: T(n,k) is the number of length-n ascent sequences without flat steps, containing k zeros.

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 2, 3, 0, 0, 0, 5, 9, 2, 0, 0, 0, 16, 32, 13, 0, 0, 0, 0, 61, 132, 72, 6, 0, 0, 0, 0, 271, 623, 409, 69, 0, 0, 0, 0, 0, 1372, 3314, 2480, 605, 24, 0, 0, 0, 0, 0, 7795, 19628, 16222, 5016, 432, 0, 0, 0, 0, 0, 0, 49093, 128126, 114594, 41955, 5498, 120, 0, 0, 0, 0, 0

Offset: 0

Author

Joerg Arndt, Nov 05 2012

Keywords

• nonn
• tabl

Comments

An ascent sequence is a sequence [d(1), d(2), ..., d(n)] where d(1)=0, d(k)>=0, and d(k) <= asc([d(1), d(2), ..., d(k-1)]) and asc(.) gives the number of ascents of its argument. Here we consider only those where adjacent digits are unequal.

The rows are the upward diagonals of A193344.

Row sums are A138265.

The column for k=1 is A138265 (i.e. the sum of row n equals the element for k=1 of the row n+1): the length-(n+1) sequences with one zero (which must be at the initial position) are formed by incrementing each digit of the length-n sequences and prepending zero.

The second column is A194530.

Examples

			Triangle starts:
1,
0, 1,
0, 1, 0,
0, 1, 1, 0,
0, 2, 3, 0, 0,
0, 5, 9, 2, 0, 0,
0, 16, 32, 13, 0, 0, 0,
0, 61, 132, 72, 6, 0, 0, 0,
0, 271, 623, 409, 69, 0, 0, 0, 0,
0, 1372, 3314, 2480, 605, 24, 0, 0, 0, 0,
0, 7795, 19628, 16222, 5016, 432, 0, 0, 0, 0, 0,
0, 49093, 128126, 114594, 41955, 5498, 120, 0, 0, 0, 0, 0,
0, 339386, 914005, 872336, 363123, 62626, 3120, 0, 0, 0, 0, 0, 0,
...
The A138265(5) = 16 length-5 ascent sequences without flat steps are (dots for zeros):
[ #]     ascent-seq.  #zeros
[ 1]    [ . 1 . 1 . ]  3
[ 2]    [ . 1 . 1 2 ]  2
[ 3]    [ . 1 . 1 3 ]  2
[ 4]    [ . 1 . 2 . ]  3
[ 5]    [ . 1 . 2 1 ]  2
[ 6]    [ . 1 . 2 3 ]  2
[ 7]    [ . 1 2 . 1 ]  2
[ 8]    [ . 1 2 . 2 ]  2
[ 9]    [ . 1 2 . 3 ]  2
[10]    [ . 1 2 1 . ]  2
[11]    [ . 1 2 1 2 ]  1
[12]    [ . 1 2 1 3 ]  1
[13]    [ . 1 2 3 . ]  2
[14]    [ . 1 2 3 1 ]  1
[15]    [ . 1 2 3 2 ]  1
[16]    [ . 1 2 3 4 ]  1
There are 5 sequences with 1 zero, 9 with two zeros and 2 with three zeros, so the row for n==5 is 0, 5, 9, 2, 0, 0.
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 0..65, flattened (rows 0..15 from Joerg Arndt)