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%I #8 Sep 15 2021 03:23:37
%S 1,0,1,2,0,1,2,0,2,4,2,0,2,3,1,4,3,2,6,4,3,5,0,5,9,3,2,7,6,3,11,10,0,
%T 9,12,3,11,10,8,11,8,9,9,6,12,19,15,7,15,16,4,20,21,10,23,24,10,16,19,
%U 18,20,20,17,24,27,18,28,26,19,33,30,12,33,39,25,36,38,16,32,44,29,41,48,37,41,45,33,39,44,41
%N Expansion of Sum_{i>=1} x^(i*(i+1)/2)/(1 + x^(i*(i+1)/2)) * Product_{j>=1} (1 + x^(j*(j+1)/2)).
%C Total number of parts in all partitions of n into distinct nonzero triangular numbers (A000217).
%H Vaclav Kotesovec, <a href="/A281666/b281666.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F G.f.: Sum_{i>=1} x^(i*(i+1)/2)/(1 + x^(i*(i+1)/2)) * Product_{j>=1} (1 + x^(j*(j+1)/2)).
%e a(10) = 4 because we have [10], [6, 3, 1] and 1 + 3 = 4.
%t nmax = 90; Rest[CoefficientList[Series[Sum[x^(i (i + 1)/2)/(1 + x^(i (i + 1)/2)), {i, 1, nmax}] Product[1 + x^(j (j + 1)/2), {j, 1, nmax}], {x, 0, nmax}], x]]
%Y Cf. A000217, A015723, A024940, A281542, A281615, A329801.
%K nonn
%O 1,4
%A _Ilya Gutkovskiy_, Jan 26 2017
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