A278340 Number of partitions of n*(n+1)/2 into distinct squares.

1, 1, 0, 0, 1, 0, 1, 0, 1, 2, 1, 3, 4, 3, 4, 4, 3, 4, 9, 14, 18, 19, 8, 16, 25, 27, 47, 37, 55, 83, 66, 92, 100, 108, 214, 189, 201, 303, 334, 535, 587, 587, 689, 764, 908, 1278, 1494, 1904, 2369, 2744, 2970, 3269, 3805, 4780, 6701, 7744, 9120, 10582, 11082

Offset: 0

Author

Alois P. Heinz, Nov 18 2016

Keywords

• nonn

Examples

			a(9) = 2: [25,16,4], [36,9].
a(10) = 1: [25,16,9,4,1].
a(11) = 3: [36,16,9,4,1], [36,25,4,1], [49,16,1].
a(12) = 4: [36,25,16,1], [49,16,9,4], [49,25,4], [64,9,4,1]
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000217, A000290, A033461, A278339.

Programs

Maple

b:= proc(n, i) option remember; (m-> `if`(n>m, 0,
      `if`(n=m, 1, b(n, i-1)+ `if`(i^2>n, 0,
         b(n-i^2, i-1)))))(i*(i+1)*(2*i+1)/6)
    end:
a:= n-> (m-> b(m, isqrt(m)))(n*(n+1)/2):
seq(a(n), n=0..80);

Mathematica

b[n_, i_] := b[n, i] = (If[n > #, 0, If[n == #, 1, b[n, i - 1] + If[i^2 > n, 0, b[n - i^2, i - 1]]]]) &[i*(i + 1)*(2*i + 1)/6];
a[n_] := b[#, Floor @ Sqrt[#]] &[n*(n + 1)/2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 20 2018, translated from Maple *)

Formula

a(n) = [x^(n*(n+1)/2)] Product_{i>=1} (1+x^(i^2)).

a(n) = A033461(A000217(n)).