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 A309730 Expansion of Sum_{k>=1} x^k * (1 - x^(3*k))/(1 - x^k)^4. 0

%I

%S 1,5,11,24,32,61,65,109,120,172,167,279,236,343,358,470,410,630,515,

%T 762,706,865,761,1193,933,1216,1174,1497,1220,1850,1397,1959,1762,

%U 2098,1882,2739,2000,2629,2470,3188,2462,3614,2711,3723,3438,3871,3245,4939,3594,4749,4246,5214

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

%C Inverse Moebius transform of centered triangular numbers (A005448).

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

%F a(n) = 3 * (sigma_2(n) - sigma_1(n))/2 + d(n).

%t nmax = 52; CoefficientList[Series[Sum[x^k (1 - x^(3 k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t Table[3 (DivisorSigma[2, n] - DivisorSigma[1, n])/2 + DivisorSigma[0, n], {n, 1, 52}]

%o (PARI) a(n)={sumdiv(n, d, 3*d*(d-1)/2 + 1)} \\ _Andrew Howroyd_, Aug 14 2019

%o (PARI) a(n)={3*(sigma(n,2) - sigma(n))/2 + numdiv(n)} \\ _Andrew Howroyd_, Aug 14 2019

%Y Cf. A000005, A000203, A001157, A005448, A007437, A059358.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Aug 14 2019

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Last modified August 11 15:12 EDT 2020. Contains 336428 sequences. (Running on oeis4.)