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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.

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