A327631 Number T(n,k) of parts in all proper k-times partitions of n; triangle T(n,k), n >= 1, 0 <= k <= n-1, read by rows.

1, 1, 2, 1, 5, 3, 1, 11, 21, 12, 1, 19, 61, 74, 30, 1, 34, 205, 461, 432, 144, 1, 53, 474, 1652, 2671, 2030, 588, 1, 85, 1246, 6795, 17487, 23133, 15262, 3984, 1, 127, 2723, 20966, 76264, 148134, 158452, 88194, 19980, 1, 191, 6277, 69812, 360114, 1002835, 1606434, 1483181, 734272, 151080

Offset

1,3

Comments

In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.

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 A327618.

Links

Alois P. Heinz, Rows n = 1..200, flattened

Wikipedia, Partition (number theory)

Formula

T(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A327618(n,i).

T(n,n-1) = n * A327639(n,n-1) = n * A327643(n) for n >= 1.

Example

T(4,0) = 1:
  4    (1 part).
T(4,1) = 11 = 2 + 2 + 3 + 4:
  4-> 31    (2 parts)
  4-> 22    (2 parts)
  4-> 211   (3 parts)
  4-> 1111  (4 parts)
T(4,2) = 21 = 3 + 4 + 3 + 3 + 4 + 4:
  4-> 31 -> 211   (3 parts)
  4-> 31 -> 1111  (4 parts)
  4-> 22 -> 112   (3 parts)
  4-> 22 -> 211   (3 parts)
  4-> 22 -> 1111  (4 parts)
  4-> 211-> 1111  (4 parts)
T(4,3) = 12 = 4 + 4 + 4:
  4-> 31 -> 211 -> 1111  (4 parts)
  4-> 22 -> 112 -> 1111  (4 parts)
  4-> 22 -> 211 -> 1111  (4 parts)
Triangle T(n,k) begins:
  1;
  1,   2;
  1,   5,    3;
  1,  11,   21,    12;
  1,  19,   61,    74,    30;
  1,  34,  205,   461,   432,    144;
  1,  53,  474,  1652,  2671,   2030,    588;
  1,  85, 1246,  6795, 17487,  23133,  15262,  3984;
  1, 127, 2723, 20966, 76264, 148134, 158452, 88194, 19980;
  ...

Maple

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
     `if`(k=0, [1, 1], `if`(i<2, 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), 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..n-1), n=1..12);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i < 2, 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], 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[T[n, k], {n, 1, 12}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Jan 07 2020, after Alois P. Heinz *)

Crossrefs

Columns k=0-2 give: A057427, -1+A006128(n), A328042.

Row sums give A327648.

T(n,floor(n/2)) gives A328041.

Cf. A327618, A327632, A327639, A327643.

Sequence in context: A210792 A105728 A120095 * A130197 A106513 A054446

Adjacent sequences: A327628 A327629 A327630 * A327632 A327633 A327634

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 19 2019

Status

approved