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A218757 Triangle read by rows: T(n,k) is the number of length-n ascent sequences without flat steps, containing k zeros. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,12

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.

LINKS

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

EXAMPLE

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.

CROSSREFS

Sequence in context: A152857 A097946 A083926 * A261430 A024466 A021817

Adjacent sequences:  A218754 A218755 A218756 * A218758 A218759 A218760

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt, Nov 05 2012

STATUS

approved

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Last modified November 23 18:56 EST 2017. Contains 295128 sequences.