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

%I A278340 #15 May 20 2018 11:35:59
%S A278340 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 A278340 83,66,92,100,108,214,189,201,303,334,535,587,587,689,764,908,1278,
%U A278340 1494,1904,2369,2744,2970,3269,3805,4780,6701,7744,9120,10582,11082
%N A278340 Number of partitions of n*(n+1)/2 into distinct squares.
%H A278340 Alois P. Heinz, <a href="/A278340/b278340.txt">Table of n, a(n) for n = 0..1000</a>
%F A278340 a(n) = [x^(n*(n+1)/2)] Product_{i>=1} (1+x^(i^2)).
%F A278340 a(n) = A033461(A000217(n)).
%e A278340 a(9) = 2: [25,16,4], [36,9].
%e A278340 a(10) = 1: [25,16,9,4,1].
%e A278340 a(11) = 3: [36,16,9,4,1], [36,25,4,1], [49,16,1].
%e A278340 a(12) = 4: [36,25,16,1], [49,16,9,4], [49,25,4], [64,9,4,1]
%p A278340 b:= proc(n, i) option remember; (m-> `if`(n>m, 0,
%p A278340       `if`(n=m, 1, b(n, i-1)+ `if`(i^2>n, 0,
%p A278340          b(n-i^2, i-1)))))(i*(i+1)*(2*i+1)/6)
%p A278340     end:
%p A278340 a:= n-> (m-> b(m, isqrt(m)))(n*(n+1)/2):
%p A278340 seq(a(n), n=0..80);
%t A278340 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 A278340 a[n_] := b[#, Floor @ Sqrt[#]] &[n*(n + 1)/2];
%t A278340 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 20 2018, translated from Maple *)
%Y A278340 Cf. A000217, A000290, A033461, A278339.
%K A278340 nonn
%O A278340 0,10
%A A278340 _Alois P. Heinz_, Nov 18 2016