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.

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

Author

Alois P. Heinz, Sep 19 2019

Keywords

• nonn
• tabf

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.

Examples

			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;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..200, flattened
• Wikipedia, Partition (number theory)

Crossrefs

Columns k=0-2 give: A057427, -1+A015723(n), A327795.

Row sums give A327647.

Cf. A327622, A327631.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
     `if`(k=0, [1, 1], `if`(i*(i+1)/2 (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 *)

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.