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A308680
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Number T(n,k) of colored integer partitions of n such that all colors from a k-set are used and parts differ by size or by color; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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15
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1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 3, 8, 9, 4, 1, 0, 4, 14, 19, 14, 5, 1, 0, 5, 22, 39, 36, 20, 6, 1, 0, 6, 34, 72, 85, 60, 27, 7, 1, 0, 8, 50, 128, 180, 160, 92, 35, 8, 1, 0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1, 0, 12, 104, 354, 680, 845, 720, 434, 184, 54, 10, 1
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OFFSET
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0,8
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COMMENTS
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For fixed k > 0, T(n,k) ~ exp(Pi*sqrt(k*n/3)) * k^(1/4) / (3^(1/4) * 2^((k+3)/2) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019
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LINKS
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FORMULA
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T(n,k) = Sum_{i=0..k} (-1)^i * binomial(k,i) * A286335(n,k-i).
Sum_{k=1..n} k * T(n,k) = A325915(n).
G.f. of column k: (-1 + Product_{j>=1} (1 + x^j))^k. - Alois P. Heinz, Jan 29 2021
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EXAMPLE
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T(4,1) = 2: 3a1a, 4a.
T(4,2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
T(4,3) = 3: 2a1b1c, 2b1a1c, 2c1a1b.
T(4,4) = 1: 1a1b1c1d.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 2, 2, 1;
0, 2, 5, 3, 1;
0, 3, 8, 9, 4, 1;
0, 4, 14, 19, 14, 5, 1;
0, 5, 22, 39, 36, 20, 6, 1;
0, 6, 34, 72, 85, 60, 27, 7, 1;
0, 8, 50, 128, 180, 160, 92, 35, 8, 1;
0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1;
...
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, 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=0..n), n=0..12);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
`if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
T:= proc(n, k) option remember;
`if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, b(n)),
(q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
# Uses function PMatrix from A357368.
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MATHEMATICA
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b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]*Binomial[k, j]][n - i*j], {j, 0, Min[k, n/i]}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, from Maple *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000009 (for n>0), A327380, A327381, A327382, A327383, A327384, A327385, A327386, A327387, A327388.
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KEYWORD
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AUTHOR
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STATUS
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approved
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