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A245840
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Triangle T read by rows: T(n,k) = Total number of odd parts in all partitions of n with exactly k parts, 1 <= k <= n.
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4
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1, 0, 2, 1, 1, 3, 0, 2, 2, 4, 1, 2, 4, 3, 5, 0, 4, 4, 6, 4, 6, 1, 3, 8, 7, 8, 5, 7, 0, 4, 8, 12, 10, 10, 6, 8, 1, 4, 13, 14, 17, 13, 12, 7, 9, 0, 6, 12, 22, 20, 22, 16, 14, 8, 10, 1, 5, 18, 25, 32, 27, 27, 19, 16, 9, 11
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OFFSET
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1,3
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins
1
0 2
1 1 3
0 2 2 4
1 2 4 3 5
0 4 4 6 4 6
1 3 8 7 8 5 7
0 4 8 12 10 10 6 8
1 4 13 14 17 13 12 7 9
0 6 12 22 20 22 16 14 8 10
1 5 18 25 32 27 27 19 16 9 11
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0, [`if`(k=0, 1, 0), 0],
`if`(i<1 or k=0, [0$2], ((f, g)-> f+g+[0, `if`(irem(i, 2)=1,
g[1], 0)])(b(n, i-1, k), `if`(i>n, [0$2], b(n-i, i, k-1)))))
end:
T:= (n, k)-> b(n$2, k)[2]:
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MATHEMATICA
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Grid[Table[Sum[Count[Flatten[IntegerPartitions[n, {k}]], i], {i, 1, n, 2}], {n, 1, 11}, {k, 1, n}]]
b[n_, i_, k_] := b[n, i, k] = If[n==0, {If[k==0, 1, 0], 0}, If[i<1 || k==0, {0, 0}, Function[{f, g}, f+g+{0, If[Mod[i, 2]==1, g[[1]], 0]}][b[n, i-1, k], If[i>n, {0, 0}, b[n-i, i, k-1]]]]]; T[n_, k_] := b[n, n, k][[2]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Nov 17 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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