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A304876 L.g.f.: log(Product_{k>=1} (1 + x^(k*(k+1)/2))) = Sum_{n>=1} a(n)*x^n/n.

%I A304876 #12 Feb 20 2023 12:26:42
%S A304876 1,-1,4,-1,1,2,1,-1,4,9,1,-10,1,-1,19,-1,1,2,1,-11,25,-1,1,-10,1,-1,4,
%T A304876 27,1,-3,1,-1,4,-1,1,26,1,-1,4,-11,1,-19,1,-1,64,-1,1,-10,1,9,4,-1,1,
%U A304876 2,56,-29,4,-1,1,-35,1,-1,25,-1,1,68,1,-1,4,9,1,-46,1,-1,19,-1,1
%N A304876 L.g.f.: log(Product_{k>=1} (1 + x^(k*(k+1)/2))) = Sum_{n>=1} a(n)*x^n/n.
%H A304876 Antti Karttunen, <a href="/A304876/b304876.txt">Table of n, a(n) for n = 1..20000</a>
%H A304876 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F A304876 G.f.: Sum_{k>=1} (k*(k + 1)/2)*x^(k*(k+1)/2)/(1 + x^(k*(k+1)/2)).
%F A304876 a(n) = Sum_{d|n} (-1)^(n/d+1)*A010054(d)*d.
%e A304876 L.g.f.: L(x) = x - x^2/2 + 4*x^3/3 - x^4/4 + x^5/5 + 2*x^6/6 + x^7/7 - x^8/8 + 4*x^9/9 + 9*x^10/10 + ...
%e A304876 exp(L(x)) = 1 + x + x^3 + x^4 + x^6 + x^7 + x^9 + 2*x^10 + x^11 + x^13 + x^14 + x^15 + 2*x^16 + ... + A024940(n)*x^n + ...
%t A304876 nmax = 77; Rest[CoefficientList[Series[Log[Product[1 + x^(k (k + 1)/2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
%t A304876 nmax = 77; Rest[CoefficientList[Series[Sum[k (k + 1)/2 x^(k (k + 1)/2)/(1 + x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]]
%t A304876 Table[DivisorSum[n, (-1)^(n/# + 1) # &, IntegerQ[(8 # + 1)^(1/2)] &], {n, 77}]
%o A304876 (PARI)
%o A304876 A010054(n) = issquare(8*n + 1);
%o A304876 A304876(n) = sumdiv(n,d,(-1)^(1+(n/d)) * A010054(d)*d); \\ _Antti Karttunen_, Feb 20 2023
%Y A304876 Cf. A000217, A010054, A024940, A185027, A300853.
%K A304876 sign
%O A304876 1,3
%A A304876 _Ilya Gutkovskiy_, May 20 2018