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%I #27 Apr 19 2021 07:25:19
%S 1,5,9,20,20,48,35,76,72,110,77,204,104,196,210,288,170,405,209,480,
%T 378,440,299,816,425,598,594,868,464,1200,527,1104,858,986,910,1800,
%U 740,1216,1170,1960,902,2184,989,1980,1890,1748,1175,3216,1470,2475,1938,2704,1484,3456,2090
%N Expansion of Sum_{k>=1} k * x^k/(1 - x^k)^3.
%C Dirichlet convolution of natural numbers (A000027) with triangular numbers (A000217).
%H Seiichi Manyama, <a href="/A309731/b309731.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f.: Sum_{k>=1} (k*(k + 1)/2) * x^k/(1 - x^k)^2.
%F a(n) = n * (d(n) + sigma(n))/2.
%F Dirichlet g.f.: zeta(s-1) * (zeta(s-2) + zeta(s-1))/2.
%F a(n) = Sum_{k=1..n} k*tau(gcd(n,k)). - _Ridouane Oudra_, Nov 28 2019
%p with(numtheory): seq(n*(tau(n)+sigma(n))/2, n=1..30); # _Ridouane Oudra_, Nov 28 2019
%t nmax = 55; CoefficientList[Series[Sum[k x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t Table[DirichletConvolve[j, j (j + 1)/2, j, n], {n, 1, 55}]
%t Table[n (DivisorSigma[0, n] + DivisorSigma[1, n])/2, {n, 1, 55}]
%o (PARI) a(n)=sumdiv(n,d,binomial(n/d+1,2)*d); \\ _Andrew Howroyd_, Aug 14 2019
%o (PARI) a(n)=n*(numdiv(n) + sigma(n))/2; \\ _Andrew Howroyd_, Aug 14 2019
%o (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+1, 2)*x^k/(1-x^k)^2)) \\ _Seiichi Manyama_, Apr 19 2021
%Y Cf. A000005, A000027, A000203, A000217, A007437, A007503, A034715, A038040, A064987, A309732.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Aug 14 2019
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