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A256193
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Number T(n,k) of partitions of n into two sorts of parts having exactly k parts of the second sort; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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14
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1, 1, 1, 2, 3, 1, 3, 6, 4, 1, 5, 12, 11, 5, 1, 7, 20, 24, 16, 6, 1, 11, 35, 49, 41, 22, 7, 1, 15, 54, 89, 91, 63, 29, 8, 1, 22, 86, 158, 186, 155, 92, 37, 9, 1, 30, 128, 262, 351, 342, 247, 129, 46, 10, 1, 42, 192, 428, 635, 700, 590, 376, 175, 56, 11, 1
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OFFSET
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0,4
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LINKS
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FORMULA
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T(n,k) = [x^k] [q^(n-k)] 1/(q+x; q)_inf = [x^k] [q^(n-k)] 1/(q+x; q)_n, where (x; q)_n is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 22 2016
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EXAMPLE
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T(3,0) = 3: 111, 21, 3.
T(3,1) = 6: 1'11, 11'1, 111', 2'1, 21', 3'.
T(3,2) = 4: 1'1'1, 1'11', 11'1', 2'1'.
T(3,3) = 1: 1'1'1'.
Triangle T(n,k) begins:
1;
1, 1;
2, 3, 1;
3, 6, 4, 1;
5, 12, 11, 5, 1;
7, 20, 24, 16, 6, 1;
11, 35, 49, 41, 22, 7, 1;
15, 54, 89, 91, 63, 29, 8, 1;
22, 86, 158, 186, 155, 92, 37, 9, 1;
30, 128, 262, 351, 342, 247, 129, 46, 10, 1;
42, 192, 428, 635, 700, 590, 376, 175, 56, 11, 1;
...
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MAPLE
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b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1)*add(x^t*
binomial(j, t), t=0..j), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);
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MATHEMATICA
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b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1]* Sum[x^t*Binomial[j, t], {t, 0, j}], {j, 0, n/i}]]]]; T[n_] := Function[ p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)
Table[SeriesCoefficient[FunctionExpand[1/QPochhammer[q + x, q, n]], {q, 0, n - k}, {x, 0, k}], {n, 0, 10}, {k, 0, n}] // Column (* Vladimir Reshetnikov, Nov 22 2016 *)
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CROSSREFS
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Column k=0-10 gives: A000041, A006128, A258472, A258473, A258474, A258475, A258476, A258477, A258478, A258479, A258480.
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KEYWORD
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AUTHOR
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STATUS
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approved
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