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A059623
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As upper right triangle, number of weakly unimodal partitions of n where initial part is k (n >= k >= 1).
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3
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1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 3, 2, 1, 1, 15, 5, 3, 2, 1, 1, 27, 8, 5, 3, 2, 1, 1, 47, 13, 7, 5, 3, 2, 1, 1, 79, 21, 11, 7, 5, 3, 2, 1, 1, 130, 33, 16, 11, 7, 5, 3, 2, 1, 1, 209, 52, 24, 15, 11, 7, 5, 3, 2, 1, 1, 330, 80, 35, 22, 15, 11, 7, 5, 3, 2, 1, 1, 512, 122, 52, 31, 22, 15, 11, 7, 5, 3
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OFFSET
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1,4
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COMMENTS
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Weakly unimodal means nondecreasing then nonincreasing.
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LINKS
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FORMULA
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T(n, k) = S(n, k) - S(n-k, k) + Sum_j[T(n-k, j)] for j >= k, where S(n, k) = A008284(n, k) = Sum_j[S(n-k, j)] for n>k >= j [note reversal] with S[n, n] = 1.
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EXAMPLE
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Rows are {1,1,2,4,8,15,...}, {1,1,2,3,5,8,...}, {1,1,2,3,5,7,...} etc.
As an upper right triangle:
1, 1, 2, 4, 8, 15, ...,
1, 1, 2, 3, 5, 8, ...,
1, 1, 2, 3, 5, 7, ...,
...
As a left downward triangle, it starts:
1;
1, 1;
2, 1, 1;
4, 2, 1, 1;
8, 3, 2, 1, 1;
15, 5, 3, 2, 1, 1;
27, 8, 5, 3, 2, 1, 1;
...
T(9,3)=11 since 9 can be written as 3+6, 3+5+1, 3+4+2, 3+4+1+1, 3+3+3, 3+3+2+1, 3+3+1+1+1, 3+2+2+2, 3+2+2+1+1, 3+2+1+1+1+1 or 3+1+1+1+1+1.
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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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