A278340 - OEIS

%I #15 May 20 2018 11:35:59

%S 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,

%T 83,66,92,100,108,214,189,201,303,334,535,587,587,689,764,908,1278,

%U 1494,1904,2369,2744,2970,3269,3805,4780,6701,7744,9120,10582,11082

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

%H Alois P. Heinz, <a href="/A278340/b278340.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%e a(9) = 2: [25,16,4], [36,9].

%e a(10) = 1: [25,16,9,4,1].

%e a(11) = 3: [36,16,9,4,1], [36,25,4,1], [49,16,1].

%e a(12) = 4: [36,25,16,1], [49,16,9,4], [49,25,4], [64,9,4,1]

%p b:= proc(n, i) option remember; (m-> `if`(n>m, 0,

%p       `if`(n=m, 1, b(n, i-1)+ `if`(i^2>n, 0,

%p          b(n-i^2, i-1)))))(i*(i+1)*(2*i+1)/6)

%p     end:

%p a:= n-> (m-> b(m, isqrt(m)))(n*(n+1)/2):

%p seq(a(n), n=0..80);

%t 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];

%t a[n_] := b[#, Floor @ Sqrt[#]] &[n*(n + 1)/2];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 20 2018, translated from Maple *)

%Y Cf. A000217, A000290, A033461, A278339.

%K nonn

%O 0,10

%A _Alois P. Heinz_, Nov 18 2016