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%I #12 Feb 22 2014 13:06:02
%S 1,2,2,3,4,4,5,7,8,8,7,13,15,16,16,11,23,29,31,32,32,15,41,55,61,63,
%T 64,64,22,72,105,119,125,127,128,128,30,127,199,233,247,253,255,256,
%U 256,42,222,378,455,489,503,509,511,512,512,56,388,716,889,967,1001,1015,1021,1023,1024,1024
%N Triangle read by rows: T(n,k) is the number of compositions of n with maximal up-step <= k; n>=1, 0<=k<n.
%C T(n,k) is the number of compositions [p(1), p(2), ..., p(k)] of n such that max(p(j) - p(j-1)) <= k.
%C Rows are partial sums of rows of A225084.
%C The first column is A000041 (partition numbers), the second column is A003116, and the third column is A224959.
%C The diagonal is A011782.
%H Joerg Arndt and Alois P. Heinz, <a href="/A225085/b225085.txt">Rows n = 1..141, flattened</a>
%e Triangle begins
%e 01: 1,
%e 02: 2, 2,
%e 03: 3, 4, 4,
%e 04: 5, 7, 8, 8,
%e 05: 7, 13, 15, 16, 16,
%e 06: 11, 23, 29, 31, 32, 32,
%e 07: 15, 41, 55, 61, 63, 64, 64,
%e 08: 22, 72, 105, 119, 125, 127, 128, 128,
%e 09: 30, 127, 199, 233, 247, 253, 255, 256, 256,
%e 10: 42, 222, 378, 455, 489, 503, 509, 511, 512, 512,
%e ...
%e The fifth row corresponds to the following statistics:
%e #: M composition
%e 01: 0 [ 1 1 1 1 1 ]
%e 02: 1 [ 1 1 1 2 ]
%e 03: 1 [ 1 1 2 1 ]
%e 04: 2 [ 1 1 3 ]
%e 05: 1 [ 1 2 1 1 ]
%e 06: 1 [ 1 2 2 ]
%e 07: 2 [ 1 3 1 ]
%e 08: 3 [ 1 4 ]
%e 09: 0 [ 2 1 1 1 ]
%e 10: 1 [ 2 1 2 ]
%e 11: 0 [ 2 2 1 ]
%e 12: 1 [ 2 3 ]
%e 13: 0 [ 3 1 1 ]
%e 14: 0 [ 3 2 ]
%e 15: 0 [ 4 1 ]
%e 16: 0 [ 5 ]
%e There are 7 compositions with no up-step (M<=0), 13 with M<=1, 15 with M<=2, 16 with M<=3, and 16 with M<=4.
%K nonn,tabl
%O 1,2
%A _Joerg Arndt_, Apr 27 2013
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