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Expansion of Sum_{k>=1} x^(k*(k + 1)/2) / (1 - x^(k*(k + 1)/2))^2.
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%I #7 Sep 19 2019 19:56:18

%S 1,2,4,4,5,9,7,8,12,11,11,18,13,14,21,16,17,27,19,22,29,22,23,36,25,

%T 26,36,29,29,50,31,32,44,34,35,55,37,38,52,44,41,65,43,44,64,46,47,72,

%U 49,55,68,52,53,81,56,58,76,58,59,100,61,62,87,64,65,100,67,68,92,77

%N Expansion of Sum_{k>=1} x^(k*(k + 1)/2) / (1 - x^(k*(k + 1)/2))^2.

%C Sum of divisors d of n such that n/d is triangular number.

%F a(n) = Sum_{d|n} A010054(n/d) * d.

%t nmax = 70; CoefficientList[Series[Sum[x^(k (k + 1)/2)/(1 - x^(k (k + 1)/2))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t a[n_] := DivisorSum[n, # &, IntegerQ[Sqrt[8 n/# + 1]] &]; Table[a[n], {n, 1, 70}]

%o (PARI) a(n)={sumdiv(n, d, if(ispolygonal(d,3), n/d))} \\ _Andrew Howroyd_, Sep 19 2019

%Y Cf. A000217, A006463, A007862, A010054, A076752, A112886 (fixed points), A185027.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Sep 19 2019