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A309730 Expansion of Sum_{k>=1} x^k * (1 - x^(3*k))/(1 - x^k)^4.

%I A309730 #13 Jan 02 2025 09:35:38
%S A309730 1,5,11,24,32,61,65,109,120,172,167,279,236,343,358,470,410,630,515,
%T A309730 762,706,865,761,1193,933,1216,1174,1497,1220,1850,1397,1959,1762,
%U A309730 2098,1882,2739,2000,2629,2470,3188,2462,3614,2711,3723,3438,3871,3245,4939,3594,4749,4246,5214
%N A309730 Expansion of Sum_{k>=1} x^k * (1 - x^(3*k))/(1 - x^k)^4.
%C A309730 Inverse Moebius transform of centered triangular numbers (A005448).
%H A309730 Amiram Eldar, <a href="/A309730/b309730.txt">Table of n, a(n) for n = 1..10000</a>
%F A309730 G.f.: Sum_{k>=1} (3*k*(k - 1)/2 + 1) * x^k/(1 - x^k).
%F A309730 a(n) = 3 * (sigma_2(n) - sigma_1(n))/2 + d(n).
%F A309730 From _Amiram Eldar_, Jan 02 2025: (Start)
%F A309730 Dirichlet g.f.: zeta(s) * (3 * zeta(s-2) - 3 * zeta(s-1) + 2 * zeta(s)) / 2.
%F A309730 Sum_{k=1..n} a(k) ~ (zeta(3)/2) * n^3. (End)
%t A309730 nmax = 52; CoefficientList[Series[Sum[x^k (1 - x^(3 k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t A309730 Table[3 (DivisorSigma[2, n] - DivisorSigma[1, n])/2 + DivisorSigma[0, n], {n, 1, 52}]
%o A309730 (PARI) a(n)={sumdiv(n, d, 3*d*(d-1)/2 + 1)} \\ _Andrew Howroyd_, Aug 14 2019
%o A309730 (PARI) a(n)={3*(sigma(n,2) - sigma(n))/2 + numdiv(n)} \\ _Andrew Howroyd_, Aug 14 2019
%Y A309730 Cf. A000005, A000203, A001157, A002117, A005448, A007437, A059358.
%K A309730 nonn,easy
%O A309730 1,2
%A A309730 _Ilya Gutkovskiy_, Aug 14 2019