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A326616
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Number T(n,k) of colored integer partitions of n using all colors of a k-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order; triangle T(n,k), n>=0, A185283(n)<=k<=n, read by rows.
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7
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1, 1, 2, 2, 5, 1, 9, 13, 9, 44, 42, 10, 96, 225, 150, 9, 152, 680, 1098, 576, 3, 155, 1350, 4155, 5201, 2266, 124, 2180, 11730, 26642, 26904, 9966, 140, 3751, 30300, 106281, 182000, 149832, 47466, 160, 6050, 69042, 348061, 896392, 1229760, 855240, 237019
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OFFSET
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0,3
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COMMENTS
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T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.
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LINKS
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Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Partition (number theory)
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FORMULA
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Sum_{k=A185283(n)..n} k * T(n,k) = A326649(n).
Sum_{n=k..A024916(k)} n * T(n,k) = A326651(k).
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EXAMPLE
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T(3,2) = 2: 2a1b, 2b1a.
T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc
Triangle T(n,k) begins:
1;
1;
2;
2, 5;
1, 9, 13;
9, 44, 42;
10, 96, 225, 150;
9, 152, 680, 1098, 576;
3, 155, 1350, 4155, 5201, 2266;
124, 2180, 11730, 26642, 26904, 9966;
140, 3751, 30300, 106281, 182000, 149832, 47466;
...
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MAPLE
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g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
h:= proc(n) option remember; local k; for k from
`if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=h(n)..n), n=0..12);
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CROSSREFS
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Main diagonal gives A178682.
Row sums give A326648.
Column sums give A326650.
Cf. A000203, A185283, A326617 (this triangle read by columns), A326649, A326651.
Sequence in context: A165922 A337293 A307834 * A249033 A068762 A326914
Adjacent sequences: A326613 A326614 A326615 * A326617 A326618 A326619
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KEYWORD
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nonn,tabf
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AUTHOR
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Alois P. Heinz, Sep 12 2019
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STATUS
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approved
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