OFFSET
0
LINKS
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
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Dec 09 2018
STATUS
approved