A341040 Number T(n,k) of partitions of n into k distinct nonzero squares; triangle T(n,k), n>=0, 0<=k<=A248509(n), read by rows.

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

Offset

0

Comments

T(n,k) is defined for n, k >= 0. The triangle contains only the terms with 0 <= k <= A248509(n). T(n,k) = 0 for k > A248509(n).

Links

Alois P. Heinz, Rows n = 0..4000, flattened

Formula

T(n,k) = [x^n*y^k] Product_{j>=1} (1 + y*x^(j^2)).

T(A000330(n),n) = 1.

Row n = [0] <=> n in { A001422 }.

Sum_{k>=0} 2^k * T(n,k) = A279360(n).

Sum_{k>=0} k * T(n,k) = A281542(n).

Sum_{k>=0} (-1)^k * T(n,k) = A276516(n).

Example

T(62,3) = 2 is the first term > 1 and counts partitions [49,9,4] and [36,25,1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0;
  0;
  0, 1;
  0, 0, 1;
  0;
  0;
  0;
  0, 1;
  0, 0, 1;
  0;
  0;
  0, 0, 1;
  0, 0, 0, 1;
  ...

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i^2>n, 0, expand(b(n-i^2, i-1)*x))))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n, isqrt(n))):
seq(T(n), n=0..45);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
b[n, i - 1] + If[i^2 > n, 0, Expand[b[n - i^2, i - 1]*x]]]];
T[n_] := CoefficientList[b[n, Floor@Sqrt[n]], x] /. {} -> {0};
T /@ Range[0, 45] // Flatten (* Jean-François Alcover, Feb 15 2021, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A000007, A010052 (for n>0), A025441, A025442, A025443, A025444, A340988, A340998, A340999, A341000, A341001.

Row sums give A033461.

Cf. A000290, A000330, A001422, A243148, A248509, A276516, A279360, A281542, A337165.

Sequence in context: A262693 A267423 A108340 * A257585 A285034 A266174

Adjacent sequences: A341037 A341038 A341039 * A341041 A341042 A341043

Keyword

nonn,look,tabf

Author

Alois P. Heinz, Feb 03 2021

Status

approved