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A321878
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Number T(n,k) of partitions of n into colored blocks of equal parts, such that all colors from a set of size k are used and the colors are introduced in increasing order; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.
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14
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1, 0, 1, 0, 2, 0, 3, 1, 0, 5, 2, 0, 7, 5, 0, 11, 9, 1, 0, 15, 17, 2, 0, 22, 28, 5, 0, 30, 47, 10, 0, 42, 74, 21, 1, 0, 56, 116, 37, 2, 0, 77, 175, 67, 5, 0, 101, 263, 112, 10, 0, 135, 385, 187, 20, 0, 176, 560, 302, 40, 1, 0, 231, 800, 479, 72, 2, 0, 297, 1135, 741, 127, 5
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OFFSET
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0,5
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COMMENTS
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T(n,k) is defined for all n>=0 and k>=0. The triangle contains only elements with 0 <= k <= A003056(n). T(n,k) = 0 for k > A003056(n).
For fixed k>=1, T(n,k) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2, 1-k))*n/3)) * sqrt(Pi^2 - 6*polylog(2, 1-k)) / (4*k!*sqrt(3*k)*Pi*n). - Vaclav Kotesovec, Sep 18 2019
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LINKS
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FORMULA
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T(n,k) = 1/k! * Sum_{i=0..k} (-1)^i*binomial(k,i) A321884(n,k-i).
T(n*(n+1)/2,n) = T(A000217(n),n) = 1.
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EXAMPLE
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T(6,1) = 11: 111111a, 2a1111a, 22a11a, 222a, 3a111a, 3a2a1a, 33a, 4a11a, 4a2a, 5a1a, 6a.
T(6,2) = 9: 2a1111b, 22a11b, 3a111b, 3a2a1b, 3a2b1a, 3a2b1b, 4a11b, 4a2b, 5a1b.
T(6,3) = 1: 3a2b1c.
Triangle T(n,k) begins:
1;
0, 1;
0, 2;
0, 3, 1;
0, 5, 2;
0, 7, 5;
0, 11, 9, 1;
0, 15, 17, 2;
0, 22, 28, 5;
0, 30, 47, 10;
0, 42, 74, 21, 1;
0, 56, 116, 37, 2;
0, 77, 175, 67, 5;
...
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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))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!:
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20);
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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[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
T[n_, k_] := Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]/k!;
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CROSSREFS
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Columns k=0-10 give: A000007, A000041 (for n>0), A327285, A327286, A327287, A327288, A327289, A327290, A327291, A327292, A327293.
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KEYWORD
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AUTHOR
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STATUS
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approved
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