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 A225084 Triangle read by rows: T(n,k) is the number of compositions of n with maximal up-step k; n>=1, 0<=k
 1, 2, 0, 3, 1, 0, 5, 2, 1, 0, 7, 6, 2, 1, 0, 11, 12, 6, 2, 1, 0, 15, 26, 14, 6, 2, 1, 0, 22, 50, 33, 14, 6, 2, 1, 0, 30, 97, 72, 34, 14, 6, 2, 1, 0, 42, 180, 156, 77, 34, 14, 6, 2, 1, 0, 56, 332, 328, 173, 78, 34, 14, 6, 2, 1, 0, 77, 600, 681, 378, 177, 78, 34, 14, 6, 2, 1, 0, 101, 1078, 1393, 818, 393, 178, 78, 34, 14, 6, 2, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. The first column is A000041 (partition numbers). Sum of first and second column is A003116. Sum of the first three columns is A224959. The second columns deviates from A054454 after the term 600. Row sums are A011782. LINKS Joerg Arndt and Alois P. Heinz, Rows n = 1..141, flattened EXAMPLE Triangle starts: 01: 1, 02: 2, 0, 03: 3, 1, 0, 04: 5, 2, 1, 0, 05: 7, 6, 2, 1, 0, 06: 11, 12, 6, 2, 1, 0, 07: 15, 26, 14, 6, 2, 1, 0, 08: 22, 50, 33, 14, 6, 2, 1, 0, 09: 30, 97, 72, 34, 14, 6, 2, 1, 0, 10: 42, 180, 156, 77, 34, 14, 6, 2, 1, 0, 11: 56, 332, 328, 173, 78, 34, 14, 6, 2, 1, 0, 12: 77, 600, 681, 378, 177, 78, 34, 14, 6, 2, 1, 0, 13: 101, 1078, 1393, 818, 393, 178, 78, 34, 14, 6, 2, 1, 0, 14: 135, 1917, 2821, 1746, 863, 397, 178, 78, 34, 14, 6, 2, 1, 0, 15: 176, 3393, 5660, 3695, 1872, 877, 398, 178, 78, 34, 14, 6, 2, 1, 0, ... The fifth row corresponds to the following statistics: #:  M   composition 01:  0  [ 1 1 1 1 1 ] 02:  1  [ 1 1 1 2 ] 03:  1  [ 1 1 2 1 ] 04:  2  [ 1 1 3 ] 05:  1  [ 1 2 1 1 ] 06:  1  [ 1 2 2 ] 07:  2  [ 1 3 1 ] 08:  3  [ 1 4 ] 09:  0  [ 2 1 1 1 ] 10:  1  [ 2 1 2 ] 11:  0  [ 2 2 1 ] 12:  1  [ 2 3 ] 13:  0  [ 3 1 1 ] 14:  0  [ 3 2 ] 15:  0  [ 4 1 ] 16:  0  [ 5 ] There are 7 compositions with no up-step (M=0), 6 with M=1, 2 with M=2, and 1 with M=3. MAPLE b:= proc(n, v) option remember; `if`(n=0, 1, add((p->       `if`(i seq(coeff(b(n, 0), x, i), i=0..n-1): seq(T(n), n=1..14);  # Alois P. Heinz, Feb 22 2014 MATHEMATICA b[n_, v_] := b[n, v] = If[n == 0, 1, Sum[Function[{p}, If[i

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Last modified October 15 03:30 EDT 2019. Contains 328025 sequences. (Running on oeis4.)