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A218757
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Triangle read by rows: T(n,k) is the number of length-n ascent sequences without flat steps, containing k zeros.
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2
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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
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OFFSET
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0,12
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COMMENTS
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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.
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LINKS
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Joerg Arndt and Alois P. Heinz, Rows n = 0..65, flattened (rows 0..15 from Joerg Arndt)
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EXAMPLE
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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.
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CROSSREFS
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Sequence in context: A152857 A097946 A083926 * A261430 A024466 A021817
Adjacent sequences: A218754 A218755 A218756 * A218758 A218759 A218760
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KEYWORD
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nonn,tabl
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AUTHOR
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Joerg Arndt, Nov 05 2012
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STATUS
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approved
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