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

0,10

Links

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

Formula

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

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

Example

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]

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 *)

Crossrefs

Cf. A000217, A000290, A033461, A278339.

Sequence in context: A358193 A122530 A301453 * A324749 A022466 A144254

Adjacent sequences: A278337 A278338 A278339 * A278341 A278342 A278343

Keyword

nonn

Author

Alois P. Heinz, Nov 18 2016

Status

approved