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Expansion of Sum_{k>0} (-1)^(k-1) * x^(3*k-1) / (1 - x^(3*k-1))^3.
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%I #24 Jul 03 2023 00:59:50

%S 0,1,0,3,-1,6,0,11,0,12,-1,21,0,29,-6,39,-1,45,0,46,0,63,-1,84,-15,92,

%T 0,108,-1,99,0,147,-6,150,-29,171,0,191,0,192,-1,237,0,244,-45,273,-1,

%U 321,0,271,-6,354,-1,378,-81,445,0,432,-1,393,0,497,0,567,-92,540,0,586,-6,537,-1,711,0,704,-120,744

%N Expansion of Sum_{k>0} (-1)^(k-1) * x^(3*k-1) / (1 - x^(3*k-1))^3.

%H Seiichi Manyama, <a href="/A364018/b364018.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>0} k*(k+1)/2 * x^(2*k) / (1 + x^(3*k)).

%F a(n) = Sum_{d|n, n/d==2 (mod 3)} (-1)^(n/d) * binomial(d+1,2).

%t a[n_] := DivisorSum[n, (-1)^(n/#) * Binomial[#+1, 2] &, Mod[n/#, 3] == 2 &]; Array[a, 100] (* _Amiram Eldar_, Jul 03 2023 *)

%o (PARI) a(n) = sumdiv(n, d, (n/d%3==2)*(-1)^(n/d)*binomial(d+1, 2));

%Y Cf. A364015, A364017.

%K sign

%O 1,4

%A _Seiichi Manyama_, Jul 01 2023