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A238344
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Irregular triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k descents, n>=0, 0<=k<=floor(n/3).
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14
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1, 1, 2, 3, 1, 5, 3, 7, 9, 11, 19, 2, 15, 41, 8, 22, 77, 29, 30, 142, 81, 3, 42, 247, 205, 18, 56, 421, 469, 78, 77, 689, 1013, 264, 5, 101, 1113, 2059, 786, 37, 135, 1750, 4021, 2097, 189, 176, 2712, 7558, 5179, 751, 8, 231, 4128, 13780, 11998, 2558, 73, 297, 6208, 24440, 26400, 7762, 429
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OFFSET
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0,3
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COMMENTS
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Same as A238343, with zeros omitted.
Columns k=0-10 give: A000041, A241626, A241627, A241628, A241629, A241630, A241631, A241632, A241633, A241634, A241635.
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LINKS
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EXAMPLE
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Triangle starts:
00: 1;
01: 1;
02: 2;
03: 3, 1;
04: 5, 3;
05: 7, 9;
06: 11, 19, 2;
07: 15, 41, 8;
08: 22, 77, 29;
09: 30, 142, 81, 3;
10: 42, 247, 205, 18;
11: 56, 421, 469, 78;
12: 77, 689, 1013, 264, 5;
13: 101, 1113, 2059, 786, 37;
14: 135, 1750, 4021, 2097, 189;
15: 176, 2712, 7558, 5179, 751, 8;
16: 231, 4128, 13780, 11998, 2558, 73;
17: 297, 6208, 24440, 26400, 7762, 429;
18: 385, 9201, 42358, 55593, 21577, 1945, 13;
19: 490, 13502, 71867, 112814, 55867, 7465, 139;
20: 627, 19585, 119715, 221639, 136478, 25317, 927;
...
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, expand(
add(b(n-j, j)*`if`(j<i, x, 1), j=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..20);
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0, 1, Expand[Sum[b[n-j, j]*If[j<i, x, 1], {j, 1, n} ]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Maple *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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