A337165 Number T(n,k) of compositions of n into k nonzero squares; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 1, 0, 0, 5, 0, 0, 1, 0, 1, 0, 3, 0, 0, 6, 0, 0, 1, 0, 0, 2, 0, 6, 0, 0, 7, 0, 0, 1, 0, 0, 0, 3, 0, 10, 0, 0, 8, 0, 0, 1, 0, 0, 0, 1, 4, 0, 15, 0, 0, 9, 0, 0, 1

Offset: 0

Author

Alois P. Heinz, Feb 03 2021

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 0, 1;
  0, 1, 0, 0, 1;
  0, 0, 2, 0, 0,  1;
  0, 0, 0, 3, 0,  0,  1;
  0, 0, 0, 0, 4,  0,  0, 1;
  0, 0, 1, 0, 0,  5,  0, 0, 1;
  0, 1, 0, 3, 0,  0,  6, 0, 0, 1;
  0, 0, 2, 0, 6,  0,  0, 7, 0, 0, 1;
  0, 0, 0, 3, 0, 10,  0, 0, 8, 0, 0, 1;
  0, 0, 0, 1, 4,  0, 15, 0, 0, 9, 0, 0, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..350, flattened

Crossrefs

Columns k=0-10 give: A000007, A010052, A063725, A063691, A063730, A340481, A340905, A340906, A340915, A340946, A340947.

Row sums give A006456.

T(2n,n) gives A338464.

Main diagonal gives A000012.

Cf. A000290, A281704, A317665, A341040.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add((s->
     `if`(s>n, 0, expand(x*b(n-s))))(j^2), j=1..isqrt(n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..14);

Mathematica

b[n_] := b[n] = If[n == 0, 1, Sum[With[{s = j^2},
     If[s>n, 0, Expand[x*b[n - s]]]], {j, 1, Sqrt[n]}]];
T[n_] := CoefficientList[b[n], x];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Feb 07 2021, after Alois P. Heinz *)

Formula

G.f. of column k: (Sum_{j>=1} x^(j^2))^k.

Sum_{k=0..n} k * T(n,k) = A281704(n).

Sum_{k=0..n} (-1)^k * T(n,k) = A317665(n).