A327618 Number A(n,k) of parts in all k-times partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 6, 1, 0, 1, 7, 14, 12, 1, 0, 1, 9, 25, 44, 20, 1, 0, 1, 11, 39, 109, 100, 35, 1, 0, 1, 13, 56, 219, 315, 274, 54, 1, 0, 1, 15, 76, 386, 769, 1179, 581, 86, 1, 0, 1, 17, 99, 622, 1596, 3643, 3234, 1417, 128, 1, 0, 1, 19, 125, 939, 2960, 9135, 12336, 10789, 2978, 192, 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 A327631.

Examples

			A(2,2) = 5 = 1+2+2 counting the parts in 2, 11, 1|1.
Square array A(n,k) begins:
  0,  0,   0,    0,     0,     0,     0,      0, ...
  1,  1,   1,    1,     1,     1,     1,      1, ...
  1,  3,   5,    7,     9,    11,    13,     15, ...
  1,  6,  14,   25,    39,    56,    76,     99, ...
  1, 12,  44,  109,   219,   386,   622,    939, ...
  1, 20, 100,  315,   769,  1596,  2960,   5055, ...
  1, 35, 274, 1179,  3643,  9135, 19844,  38823, ...
  1, 54, 581, 3234, 12336, 36911, 93302, 208377, ...
		

Links

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

Crossrefs

Columns k=0-3 give: A057427, A006128, A327594, A327627.

Rows n=0-3 give: A000004, A000012, A005408, A095794(k+1).

Main diagonal gives A327619.

Cf. A323718, A327622, A327631.

Programs

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:
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] = 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]]]];
A[n_, k_] := b[n, n, k][[2]];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

Formula

A(n,k) = Sum_{i=0..k} binomial(k,i) * A327631(n,i).