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A238406
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Number T(n,k) of partitions of n into k parts such that every i-th smallest part (counted with multiplicity) is different from i; triangle T(n,k), n>=0, 0<=k<=floor((sqrt(9+8*n)-3)/2) read by rows.
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10
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1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 3, 0, 1, 3, 1, 0, 1, 4, 3, 0, 1, 4, 4, 0, 1, 5, 6, 0, 1, 5, 7, 0, 1, 6, 9, 1, 0, 1, 6, 11, 4, 0, 1, 7, 13, 7, 0, 1, 7, 15, 11, 0, 1, 8, 18, 15, 0, 1, 8, 20, 19, 0, 1, 9, 23, 25, 1, 0, 1, 9, 26, 30, 5
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OFFSET
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0,14
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LINKS
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EXAMPLE
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T(10,1) = 1: [10].
T(10,2) = 4: [5,5], [4,6], [3,7], [2,8].
T(10,3) = 3: [3,3,4], [2,4,4], [2,3,5].
Triangle T(n,k) begins:
1;
0;
0, 1;
0, 1;
0, 1;
0, 1, 1;
0, 1, 2;
0, 1, 2;
0, 1, 3;
0, 1, 3, 1;
0, 1, 4, 3;
0, 1, 4, 4;
0, 1, 5, 6;
0, 1, 5, 7;
0, 1, 6, 9, 1;
...
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, (p-> expand(
x*(p-coeff(p, x, i-1)*x^(i-1))))(b(n-i, i)))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n$2)):
seq(T(n), n=0..30);
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[p, Expand[x*(p - Coefficient[p, x, i-1]*x^(i-1))]][b[n-i, i]]]] ]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Max[0, Exponent[p, x]]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000012 (for n>1), A004526(n-2) (for n>4), A244239, A244240, A244241, A244242, A244243, A244244, A244245, A244246.
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KEYWORD
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AUTHOR
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STATUS
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approved
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