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Expansion of Sum_{k>=1} k * x^k / (1 - x^(2*k))^2.
7

%I #30 Sep 08 2022 08:46:24

%S 1,2,5,4,8,10,11,8,20,16,17,20,20,22,42,16,26,40,29,32,58,34,35,40,53,

%T 40,74,44,44,84,47,32,90,52,94,80,56,58,106,64,62,116,65,68,174,70,71,

%U 80,102,106,138,80,80,148,146,88,154,88,89,168,92,94,241,64,172

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

%H Antti Karttunen, <a href="/A328203/b328203.txt">Table of n, a(n) for n = 1..20000</a>

%H Jon Maiga, <a href="http://sequencedb.net/s/A328203">Computer-generated formulas for A328203</a>, Sequence Machine.

%F a(n) = (n * d(n) + sigma(n)) / 2 if n odd, (n * (d(n) - d(n/2)) + sigma(n) - sigma(n/2)) / 2 if n even.

%F a(n) = (n * A001227(n) + A002131(n)) / 2.

%F a(2*n) = 2 * a(n).

%F From _Antti Karttunen_, Nov 13 2021: (Start)

%F The following two convolutions were found by Jon Maiga's Sequence Machine search algorithm. Both are easy to prove:

%F a(n) = Sum_{d|n} A003602(d) * A026741(n/d).

%F a(n) = Sum_{d|n} A109168(d) * A193356(n/d), where A109168(d) = A140472(d) = (d+A006519(d))/2.

%F (End)

%t nmax = 65; CoefficientList[Series[Sum[k x^k/(1 - x^(2 k))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t a[n_] := DivisorSum[n, (n Mod[#, 2] + Boole[OddQ[n/#]] #)/2 &]; Table[a[n], {n, 1, 65}]

%o (Magma) a:=[]; for k in [1..65] do if IsOdd(k) then a[k]:=(k * #Divisors(k) + DivisorSigma(1,k)) / 2; else a[k]:=(k * (#Divisors(k) - #Divisors(k div 2)) + DivisorSigma(1,k) - DivisorSigma(1,k div 2)) / 2; end if; end for; a; // _Marius A. Burtea_, Oct 07 2019

%o (PARI) A328203(n) = if(n%2,(1/2)*(sigma(n)+(n*numdiv(n))),2*A328203(n/2)); \\ _Antti Karttunen_, Nov 13 2021

%Y Cf. A000005, A000079 (fixed points), A000203, A001227, A002131, A003602, A006519, A006918, A026741, A038040, A109168, A113415, A140472, A152211, A193356, A245579, A349353 (Dirichlet inverse), A349354.

%Y Cf. also A347957.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Oct 07 2019