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A326962
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Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=k*2^(k-1), read by columns.
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9
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1, 1, 2, 2, 1, 5, 12, 18, 20, 18, 15, 11, 6, 3, 1, 15, 64, 166, 332, 566, 864, 1214, 1596, 1975, 2320, 2600, 2780, 2842, 2780, 2600, 2320, 1979, 1608, 1238, 908, 626, 404, 246, 136, 69, 32, 12, 4, 1, 52, 340, 1315, 3895, 9770, 21848, 44880, 86275, 157140
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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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FORMULA
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Sum_{k=1..n} k * T(n,k) = A327115(n).
T(n*2^(n-1),n) = T(A001787(n),n) = 1.
T(n*2^(n-1)-1,n) = n for n >= 2.
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EXAMPLE
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T(4,3) = 12: 3abc1a, 3abc1b, 3abc1c, 2ab2ac, 2ab2bc, 2ac2bc, 2ab1a1c, 2ab1b1c, 2ac1a1b, 2ac1b1c, 2bc1a1b, 2bc1a1c.
Triangle T(n,k) begins:
1;
1;
2;
2, 5;
1, 12, 15;
18, 64, 52;
20, 166, 340, 203;
18, 332, 1315, 1866, 877;
15, 566, 3895, 9930, 10710, 4140;
11, 864, 9770, 39960, 74438, 64520, 21147;
6, 1214, 21848, 134871, 386589, 564508, 408096, 115975;
...
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MAPLE
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C:= binomial:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i), j), j=0..n/i)))
end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), n=k..k*2^(k-1)), k=0..5);
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MATHEMATICA
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c = Binomial;
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1], k] c[c[k, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k-i) c[k, i], {i, 0, k}];
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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