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A325433
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Triangle T read by rows: T(n,k) is the number of partitions of n in which k is the least integer that is not a part and there are more parts > k than there are < k (n >= k > 0).
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4
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0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 8, 2, 0, 0, 0, 0, 0, 0, 0, 12, 3, 0, 0, 0, 0, 0, 0, 0, 0, 14, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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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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T(n,k) = (-1)^(k-1)*Sum_{j=0..k-1} (-1)^j*(p(n - j*(3*j + 1)/2) - p(n - j*(3*j + 5)/2 - 1)), where p(n) = A000041(n) is the number of partitions of n (see Theorem 1.1 in Andrews and Merca).
1st column: T(n,1) = A002865(n) for n > 0.
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EXAMPLE
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T(9,2) = 3 from 6 + 3 = 5 + 3 + 1 = 4 + 4 + 1 = 3 + 3 + 3.
The triangle T(n, k) begins
n\k| 1 2 3 4 5 6 7 8 9
---+------------------------------------
1 | 0
2 | 1 0
3 | 1 0 0
4 | 2 0 0 0
5 | 2 0 0 0 0
6 | 4 0 0 0 0 0
7 | 4 1 0 0 0 0 0
8 | 7 1 0 0 0 0 0 0
9 | 8 2 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*(3*j+1)/2]-PartitionsP[n-j*(3*j+5)/2-1]), {j, 0, k-1}]; Flatten[Table[T[n, k], {n, 1, 15}, {k, 1, n}]]
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PROG
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(PARI)
T(n, k) = (-1)^(k-1)*sum(j=0, k-1, (-1)^j*(numbpart(n-j*(3*j+1)/2)-numbpart(n-j*(3*j+5)/2-1)));
tabl(nn) = for(n=1, nn, for(k=1, n, print1(T(n, k), ", ")); print);
tabl(10) \\ yields sequence in triangular form
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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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