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A364891
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Triangle read by rows: T(n,k) = (-1)^(k-1)*Sum_{j=0..k-1} (-1)^j*(p(n - j*(2*j + 1)) - p(n - (j + 1)*(2*j + 1))), where p(n) = A000041(n) is the number of partitions of n, and 1 <= k <= n.
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3
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0, 1, -1, 1, 0, 0, 2, -1, 1, -1, 2, 0, 0, 0, 0, 4, -2, 2, -2, 2, -2, 4, 0, 0, 0, 0, 0, 0, 7, -2, 2, -2, 2, -2, 2, -2, 8, 0, 0, 0, 0, 0, 0, 0, 0, 12, -2, 3, -3, 3, -3, 3, -3, 3, -3, 14, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, -2, 4, -4, 4, -4, 4, -4, 4, -4, 4, -4
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OFFSET
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1,7
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LINKS
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FORMULA
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1st column: T(n,1) = A002865(n) for n > 0.
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EXAMPLE
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The triangle begins:
0;
1, -1;
1, 0, 0;
2, -1, 1, -1;
2, 0, 0, 0, 0;
4, -2, 2, -2, 2, -2;
4, 0, 0, 0, 0, 0, 0;
7, -2, 2, -2, 2, -2, 2, -2;
8, 0, 0, 0, 0, 0, 0, 0, 0;
12, -2, 3, -3, 3, -3, 3, -3, 3, -3;
14, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0;
...
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MATHEMATICA
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T[n_, k_]:=(-1)^(k-1)*Sum[(-1)^j*(PartitionsP[n-j(2j+1)]-PartitionsP[n-(j+1)(2j+1)]), {j, 0, k-1}]; Flatten[Table[T[n, k], {n, 1, 12}, {k, 1, n}]]
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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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