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A242086
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Triangle read by rows: T(n,k) is the number of compositions of n into odd parts with first part k.
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0
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1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 3, 0, 1, 0, 1, 0, 5, 0, 2, 0, 1, 0, 0, 8, 0, 3, 0, 1, 0, 1, 0, 13, 0, 5, 0, 2, 0, 1, 0, 0, 21, 0, 8, 0, 3, 0, 1, 0, 1, 0, 34, 0, 13, 0, 5, 0, 2, 0, 1, 0, 0, 55, 0, 21, 0, 8, 0, 3, 0, 1, 0, 1, 0, 89, 0, 34, 0, 13, 0, 5, 0, 2, 0, 1, 0, 0, 144, 0, 55, 0, 21, 0, 8, 0, 3, 0, 1, 0, 1
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OFFSET
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0,12
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LINKS
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FORMULA
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T(0,0)=1, T(0,k)=0 for k != 1 (first column).
T(n,k) = 0 for k>=1 and even.
T(n,n) = 1 for n>=1 and odd, otherwise T(n,n)=0.
T(n,k) = F(n-k-1) for n>=1 and odd k>=1, F = A000045.
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EXAMPLE
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Triangle starts:
00: 1,
01: 0, 1,
02: 0, 1, 0,
03: 0, 1, 0, 1,
04: 0, 2, 0, 1, 0,
05: 0, 3, 0, 1, 0, 1,
06: 0, 5, 0, 2, 0, 1, 0,
07: 0, 8, 0, 3, 0, 1, 0, 1,
08: 0, 13, 0, 5, 0, 2, 0, 1, 0,
09: 0, 21, 0, 8, 0, 3, 0, 1, 0, 1,
10: 0, 34, 0, 13, 0, 5, 0, 2, 0, 1, 0,
11: 0, 55, 0, 21, 0, 8, 0, 3, 0, 1, 0, 1,
12: 0, 89, 0, 34, 0, 13, 0, 5, 0, 2, 0, 1, 0,
13: 0, 144, 0, 55, 0, 21, 0, 8, 0, 3, 0, 1, 0, 1,
14: 0, 233, 0, 89, 0, 34, 0, 13, 0, 5, 0, 2, 0, 1, 0,
15: 0, 377, 0, 144, 0, 55, 0, 21, 0, 8, 0, 3, 0, 1, 0, 1,
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MAPLE
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b:= proc(n) b(n):=`if`(n=0, 1, add(b(n-2*j-1), j=0..(n-1)/2)) end:
T:= (n, k)-> `if`([n, k]=[0$2], 1, `if`(irem(k, 2)=0 or k>n, 0, b(n-k))):
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MATHEMATICA
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b[n_] := If[n == 0, 1, Sum[b[n-2*j-1], {j, 0, (n-1)/2}]]; T[n_, k_] := If[{n, k} == {0, 0}, 1, If[Mod[k, 2] == 0 || k>n, 0, b[n-k]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 15}] // Flatten (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
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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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