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A327632
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.
5
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
OFFSET
1,4
COMMENTS
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.
LINKS
FORMULA
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.
EXAMPLE
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;
...
MAPLE
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);
MATHEMATICA
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 *)
CROSSREFS
Columns k=0-2 give: A057427, -1+A015723(n), A327795.
Row sums give A327647.
Sequence in context: A210790 A224653 A101391 * A117704 A078032 A162453
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Sep 19 2019
STATUS
approved