login
Expansion of Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) * Product_{j>=i} 1/(1 - x^(j^2)).
1

%I #10 Apr 05 2017 04:46:57

%S 1,2,3,5,6,8,10,14,16,19,23,30,33,38,44,55,60,69,77,93,102,113,126,

%T 148,162,177,198,226,246,268,293,334,361,392,424,480,516,556,601,668,

%U 721,773,835,917,990,1054,1129,1239,1325,1415,1508,1649,1757,1875,1990,2157,2303,2441,2595,2796

%N Expansion of Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) * Product_{j>=i} 1/(1 - x^(j^2)).

%C Total number of smallest parts in all partitions of n into squares (A000290).

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%F G.f.: Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) * Product_{j>=i} 1/(1 - x^(j^2)).

%e a(9) = 16 because we have [9], [4, 4, 1], [4, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1] and 1 + 1 + 5 + 9 = 16.

%t nmax = 60; Rest[CoefficientList[Series[Sum[x^i^2/(1 - x^i^2) Product[1/(1 - x^j^2), {j, i, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]]

%o (PARI) x='x+O('x^61); Vec(sum(i=1, 60, x^i^2/(1 - x^i^2) * prod(j=i, 60, 1/(1 - x^j^2)))) \\ _Indranil Ghosh_, Apr 04 2017

%Y Cf. A000290, A001156, A092268, A092269, A195820, A281541.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Apr 03 2017