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

0,9

Comments

Row n is binomial transform of the n-th row of triangle A327631.

Links

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Wikipedia, Partition (number theory)

Formula

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

Example

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, ...

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 *)

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.

Sequence in context: A285574 A354821 A081719 * A121314 A119271 A364310

Adjacent sequences: A327615 A327616 A327617 * A327619 A327620 A327621

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 19 2019

Status

approved