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%I #22 Mar 24 2017 00:47:54
%S 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,
%T 72,6,0,0,0,0,271,623,409,69,0,0,0,0,0,1372,3314,2480,605,24,0,0,0,0,
%U 0,7795,19628,16222,5016,432,0,0,0,0,0,0,49093,128126,114594,41955,5498,120,0,0,0,0,0
%N Triangle read by rows: T(n,k) is the number of length-n ascent sequences without flat steps, containing k zeros.
%C 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.
%C The rows are the upward diagonals of A193344.
%C Row sums are A138265.
%C 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.
%C The second column is A194530.
%H Joerg Arndt and Alois P. Heinz, <a href="/A218757/b218757.txt">Rows n = 0..65, flattened</a> (rows 0..15 from Joerg Arndt)
%e Triangle starts:
%e 1,
%e 0, 1,
%e 0, 1, 0,
%e 0, 1, 1, 0,
%e 0, 2, 3, 0, 0,
%e 0, 5, 9, 2, 0, 0,
%e 0, 16, 32, 13, 0, 0, 0,
%e 0, 61, 132, 72, 6, 0, 0, 0,
%e 0, 271, 623, 409, 69, 0, 0, 0, 0,
%e 0, 1372, 3314, 2480, 605, 24, 0, 0, 0, 0,
%e 0, 7795, 19628, 16222, 5016, 432, 0, 0, 0, 0, 0,
%e 0, 49093, 128126, 114594, 41955, 5498, 120, 0, 0, 0, 0, 0,
%e 0, 339386, 914005, 872336, 363123, 62626, 3120, 0, 0, 0, 0, 0, 0,
%e ...
%e The A138265(5) = 16 length-5 ascent sequences without flat steps are (dots for zeros):
%e [ #] ascent-seq. #zeros
%e [ 1] [ . 1 . 1 . ] 3
%e [ 2] [ . 1 . 1 2 ] 2
%e [ 3] [ . 1 . 1 3 ] 2
%e [ 4] [ . 1 . 2 . ] 3
%e [ 5] [ . 1 . 2 1 ] 2
%e [ 6] [ . 1 . 2 3 ] 2
%e [ 7] [ . 1 2 . 1 ] 2
%e [ 8] [ . 1 2 . 2 ] 2
%e [ 9] [ . 1 2 . 3 ] 2
%e [10] [ . 1 2 1 . ] 2
%e [11] [ . 1 2 1 2 ] 1
%e [12] [ . 1 2 1 3 ] 1
%e [13] [ . 1 2 3 . ] 2
%e [14] [ . 1 2 3 1 ] 1
%e [15] [ . 1 2 3 2 ] 1
%e [16] [ . 1 2 3 4 ] 1
%e 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.
%K nonn,tabl
%O 0,12
%A _Joerg Arndt_, Nov 05 2012
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