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A238130
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Triangle read by rows: T(n,k) is the number of compositions into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=n.
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29
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1, 1, 0, 2, 0, 0, 2, 2, 0, 0, 3, 4, 1, 0, 0, 2, 10, 4, 0, 0, 0, 4, 12, 14, 2, 0, 0, 0, 2, 22, 29, 10, 1, 0, 0, 0, 4, 26, 56, 36, 6, 0, 0, 0, 0, 3, 34, 100, 86, 31, 2, 0, 0, 0, 0, 4, 44, 148, 200, 99, 16, 1, 0, 0, 0, 0, 2, 54, 230, 374, 278, 78, 8, 0, 0, 0, 0, 0, 6, 58, 322, 680, 654, 274, 52, 2, 0, 0, 0, 0, 0
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OFFSET
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0,4
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COMMENTS
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LINKS
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EXAMPLE
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Triangle starts:
00: 1,
01: 1, 0,
02: 2, 0, 0,
03: 2, 2, 0, 0,
04: 3, 4, 1, 0, 0,
05: 2, 10, 4, 0, 0, 0,
06: 4, 12, 14, 2, 0, 0, 0,
07: 2, 22, 29, 10, 1, 0, 0, 0,
08: 4, 26, 56, 36, 6, 0, 0, 0, 0,
09: 3, 34, 100, 86, 31, 2, 0, 0, 0, 0,
10: 4, 44, 148, 200, 99, 16, 1, 0, 0, 0, 0,
11: 2, 54, 230, 374, 278, 78, 8, 0, 0, 0, 0, 0,
12: 6, 58, 322, 680, 654, 274, 52, 2, 0, 0, 0, 0, 0,
13: 2, 74, 446, 1122, 1390, 814, 225, 22, 1, 0, 0, 0, 0, 0,
...
Row 5 is [2, 10, 4, 0, 0, 0] because in the 16 compositions of 5
##: [composition] no. of changes
01: [ 1 1 1 1 1 ] 0
02: [ 1 1 1 2 ] 1
03: [ 1 1 2 1 ] 2
04: [ 1 1 3 ] 1
05: [ 1 2 1 1 ] 2
06: [ 1 2 2 ] 1
07: [ 1 3 1 ] 2
08: [ 1 4 ] 1
09: [ 2 1 1 1 ] 1
10: [ 2 1 2 ] 2
11: [ 2 2 1 ] 1
12: [ 2 3 ] 1
13: [ 3 1 1 ] 1
14: [ 3 2 ] 1
15: [ 4 1 ] 1
16: [ 5 ] 0
there are 2 with no changes, 10 with one change, and 4 with two changes.
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MAPLE
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b:= proc(n, v) option remember; `if`(n=0, 1, expand(
add(b(n-i, i)*`if`(v=0 or v=i, 1, x), i=1..n)))
end:
T:= n-> seq(coeff(b(n, 0), x, i), i=0..n):
seq(T(n), n=0..14);
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MATHEMATICA
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b[n_, v_] := b[n, v] = If[n == 0, 1, Sum[b[n-i, i]*If[v == 0 || v == i, 1, x], {i, 1, n}]]; T[n_] := Table[Coefficient[b[n, 0], x, i], {i, 0, n}]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 12 2015, translated from Maple *)
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CROSSREFS
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Cf. A238279 (same sequence with zeros omitted).
Cf. A106356 (compositions with k successive parts same).
Cf. A225084 (compositions with maximal up-step k).
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KEYWORD
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AUTHOR
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STATUS
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approved
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