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 A238343 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k descents, n>=0, 0<=k<=n. 28
 1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 5, 3, 0, 0, 0, 7, 9, 0, 0, 0, 0, 11, 19, 2, 0, 0, 0, 0, 15, 41, 8, 0, 0, 0, 0, 0, 22, 77, 29, 0, 0, 0, 0, 0, 0, 30, 142, 81, 3, 0, 0, 0, 0, 0, 0, 42, 247, 205, 18, 0, 0, 0, 0, 0, 0, 0, 56, 421, 469, 78, 0, 0, 0, 0, 0, 0, 0, 0, 77, 689, 1013, 264, 5, 0, 0, 0, 0, 0, 0, 0, 0, 101, 1113, 2059, 786, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Counting ascents gives the same triangle. For n > 0, also the number of compositions of n with k + 1 maximal weakly increasing runs. - Gus Wiseman, Mar 23 2020 LINKS Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened EXAMPLE Triangle starts: 00: 1; 01: 1, 0; 02: 2, 0, 0; 03: 3, 1, 0, 0; 04: 5, 3, 0, 0, 0; 05: 7, 9, 0, 0, 0, 0; 06: 11, 19, 2, 0, 0, 0, 0; 07: 15, 41, 8, 0, 0, 0, 0, 0; 08: 22, 77, 29, 0, 0, 0, 0, 0, 0; 09: 30, 142, 81, 3, 0, 0, 0, 0, 0, 0; 10: 42, 247, 205, 18, 0, 0, 0, 0, 0, 0, 0; 11: 56, 421, 469, 78, 0, 0, 0, 0, 0, 0, 0, 0; 12: 77, 689, 1013, 264, 5, 0, 0, 0, 0, 0, 0, 0, 0; 13: 101, 1113, 2059, 786, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0; 14: 135, 1750, 4021, 2097, 189, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0; 15: 176, 2712, 7558, 5179, 751, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0; ... From Gus Wiseman, Mar 23 2020: (Start) Row n = 5 counts the following compositions: (5) (3,2) (1,4) (4,1) (2,3) (1,3,1) (1,1,3) (2,1,2) (1,2,2) (2,2,1) (1,1,1,2) (3,1,1) (1,1,1,1,1) (1,1,2,1) (1,2,1,1) (2,1,1,1) (End) MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, expand( add(b(n-j, j)*`if`(j (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)): seq(T(n), n=0..20); MATHEMATICA b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n-j, j]*If[j

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Last modified September 30 02:50 EDT 2023. Contains 365781 sequences. (Running on oeis4.)