A320841 Number T(n,k) of partitions of n into k positive cubes; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset

0

Links

Alois P. Heinz, Rows n = 0..500, flattened

Index entries for sequences related to sums of cubes

Formula

T(n,k) = [x^n y^k] 1/Product_{j>=1} (1 - y*x^(j^3)).

Example

Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 0, 1;
  0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 1;
  0, 1, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 1, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;
  ...

Maple

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(i^3>n, 0, b(n-i^3, i, t-1))))
    end:
T:= (n, k)-> b(n, iroot(n, 3), k):
seq(seq(T(n, k), k=0..n), n=0..16);  # Alois P. Heinz, Dec 21 2018

Mathematica

T[n_, k_] := Count[PowersRepresentations[n, k, 3], r_ /; FreeQ[r, 0]]; T[0, 0] = 1; Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] + If[i^3 > n, 0, b[n - i^3, i, t - 1]]]];
T[n_, k_] := b[n, n^(1/3) // Floor, k];
Table[T[n, k], {n, 0, 16}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 08 2020, after Alois P. Heinz *)

Crossrefs

Columns k=0..10 give A000007, A010057 (for n > 0), A025455, A025456, A025457, A025458, A025459, A025460, A025461, A025462, A025463.

Row sums give A003108.

Cf. A000578, A243148, A319797.

Sequence in context: A363800 A134286 A023531 * A351725 A243148 A089495

Adjacent sequences: A320838 A320839 A320840 * A320842 A320843 A320844

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Dec 09 2018

Status

approved