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A327632
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Number T(n,k) of parts in all proper k-times partitions of n into distinct parts; triangle T(n,k), n >= 1, 0 <= k <= max(0,n-2), read by rows.
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5
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1, 1, 1, 2, 1, 2, 3, 1, 4, 6, 4, 1, 7, 13, 12, 5, 1, 9, 30, 52, 35, 6, 1, 12, 61, 137, 156, 72, 7, 1, 17, 121, 384, 638, 548, 196, 8, 1, 24, 210, 880, 1983, 2442, 1543, 400, 9, 1, 29, 353, 2012, 6211, 10865, 10555, 5231, 1026, 10, 1, 39, 600, 4477, 17883, 40855, 54279, 40511, 15178, 2070, 11
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OFFSET
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1,4
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COMMENTS
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In each step at least one part is replaced by the partition of itself into smaller distinct parts. The parts are not resorted and the parts in the result are not necessarily distinct.
T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.
Row n is the inverse binomial transform of the n-th row of array A327622.
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LINKS
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Alois P. Heinz, Rows n = 1..200, flattened
Wikipedia, Partition (number theory)
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FORMULA
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T(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A327622(n,i).
T(n+1,n-1) = 1 for n >= 1.
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EXAMPLE
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T(4,0) = 1:
4 (1 part).
T(4,1) = 2:
4-> 31 (2 parts)
T(4,2) = 3:
4-> 31 -> 211 (3 parts)
Triangle T(n,k) begins:
1;
1;
1, 2;
1, 2, 3;
1, 4, 6, 4;
1, 7, 13, 12, 5;
1, 9, 30, 52, 35, 6;
1, 12, 61, 137, 156, 72, 7;
1, 17, 121, 384, 638, 548, 196, 8;
1, 24, 210, 880, 1983, 2442, 1543, 400, 9;
1, 29, 353, 2012, 6211, 10865, 10555, 5231, 1026, 10;
...
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
`if`(k=0, [1, 1], `if`(i*(i+1)/2<n, 0, b(n, i-1, k)+
(h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))
end:
T:= (n, k)-> add(b(n$2, i)[2]*(-1)^(k-i)*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..max(0, n-2)), n=1..14);
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MATHEMATICA
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b[n_, i_, k_] := b[n, i, k] = With[{}, If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i (i + 1)/2 < n, {0, 0}, b[n, i - 1, k] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/h[[1]]}][h[[1]] b[n - i, Min[n - i, i - 1], k]]][ b[i, i, k - 1]]]]]];
T[n_, k_] := Sum[b[n, n, i][[2]] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, 0, Max[0, n - 2]}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)
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CROSSREFS
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Columns k=0-2 give: A057427, -1+A015723(n), A327795.
Row sums give A327647.
Cf. A327622, A327631.
Sequence in context: A210790 A224653 A101391 * A117704 A078032 A162453
Adjacent sequences: A327629 A327630 A327631 * A327633 A327634 A327635
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KEYWORD
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nonn,tabf
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AUTHOR
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Alois P. Heinz, Sep 19 2019
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STATUS
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approved
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