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A078616 a(n) = Sum_{k=0..n} A010815(k).

%I A078616 #49 Sep 26 2018 15:45:35
%S A078616 1,0,-1,-1,-1,0,0,1,1,1,1,1,0,0,0,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,
%T A078616 1,1,1,1,1,1,0,0,0,0,0,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,1,
%U A078616 1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1
%N A078616 a(n) = Sum_{k=0..n} A010815(k).
%C A078616 To construct the sequence: a(0)=1, a(1)=0, then (2*1+1) (-1)'s followed by 2 0's, followed by (2*2+1) 1's, followed by 3 0's, followed by (2*3+1) (-1)'s, etc.
%C A078616 From _George Beck_, May 05 2017: (Start)
%C A078616 a(n) = (Number of ones in the distinct partitions of n with an odd number of parts) - (number of ones in the distinct partitions of n with an even number of parts) (conjectured).
%C A078616 The partial sums give A246575. (End) [corrected by _Ilya Gutkovskiy_, Aug 18 2018]
%H A078616 Seiichi Manyama, <a href="/A078616/b078616.txt">Table of n, a(n) for n = 0..10000</a>
%H A078616 Mircea Merca, <a href="https://www.researchgate.net/publication/322245883_Higher-order_differences_and_higher-order_partial_sums_of_Euler%27s_partition_function">Higher-order differences and higher-order partial sums of Euler's partition function</a>, 2018.
%F A078616 For m > 0, a(k)=0 if A000326(m) <= k < A000326(m) + m; a(k)=(-1)^m if A000326(m) + m <= k < A000326(m+1).
%F A078616 G.f.: eta(x)/(1-x). - _Benoit Cloitre_, Jan 31 2004
%F A078616 G.f.: exp(-Sum_{k>=1} (sigma_1(k) - 1)*x^k/k). - _Ilya Gutkovskiy_, Aug 18 2018
%o A078616 (PARI) a(n)=polcoeff(eta(x)/(1-x)+O(x^n),n)
%Y A078616 Cf. A010815, A000326, A246575.
%K A078616 sign
%O A078616 0,1
%A A078616 _Benoit Cloitre_, Dec 10 2002