A327622 Number A(n,k) of parts in all k-times partitions of n into distinct parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 5, 3, 1, 0, 1, 1, 7, 8, 5, 1, 0, 1, 1, 9, 16, 15, 8, 1, 0, 1, 1, 11, 27, 35, 28, 10, 1, 0, 1, 1, 13, 41, 69, 73, 49, 13, 1, 0, 1, 1, 15, 58, 121, 160, 170, 86, 18, 1, 0, 1, 1, 17, 78, 195, 311, 460, 357, 156, 25, 1

Offset: 0

Author

Alois P. Heinz, Sep 19 2019

Keywords

• nonn
• tabl

Comments

Row n is binomial transform of the n-th row of triangle A327632.

Examples

			Square array A(n,k) begins:
  0,  0,  0,   0,    0,    0,    0,     0,     0, ...
  1,  1,  1,   1,    1,    1,    1,     1,     1, ...
  1,  1,  1,   1,    1,    1,    1,     1,     1, ...
  1,  3,  5,   7,    9,   11,   13,    15,    17, ...
  1,  3,  8,  16,   27,   41,   58,    78,   101, ...
  1,  5, 15,  35,   69,  121,  195,   295,   425, ...
  1,  8, 28,  73,  160,  311,  553,   918,  1443, ...
  1, 10, 49, 170,  460, 1047, 2106,  3865,  6611, ...
  1, 13, 86, 357, 1119, 2893, 6507, 13182, 24625, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..200, flattened
• Wikipedia, Partition (number theory)

Crossrefs

Columns k=0-3 give: A057427, A015723, A327605, A327628.

Rows n=0,(1+2),3-5 give: A000004, A000012, A005408, A104249, A005894.

Main diagonal gives: A327623.

Cf. A327618, A327632.

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:
A:= (n, k)-> b(n$2, k)[2]:
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

b[n_, i_, k_] := b[n, i, k] = With[{}, If[n==0, Return@{1, 0}]; If[k == 0, Return@{1, 1}]; If[i(i + 1)/2 < n, Return@{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]]];
A[n_, k_] := b[n, n, k][[2]];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2020, after Maple *)

Formula

A(n,k) = Sum_{i=0..k} binomial(k,i) * A327632(n,i).