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A328203
Expansion of Sum_{k>=1} k * x^k / (1 - x^(2*k))^2.
7
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, 40, 74, 44, 44, 84, 47, 32, 90, 52, 94, 80, 56, 58, 106, 64, 62, 116, 65, 68, 174, 70, 71, 80, 102, 106, 138, 80, 80, 148, 146, 88, 154, 88, 89, 168, 92, 94, 241, 64, 172
OFFSET
1,2
LINKS
Jon Maiga, Computer-generated formulas for A328203, Sequence Machine.
FORMULA
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.
a(n) = (n * A001227(n) + A002131(n)) / 2.
a(2*n) = 2 * a(n).
From Antti Karttunen, Nov 13 2021: (Start)
The following two convolutions were found by Jon Maiga's Sequence Machine search algorithm. Both are easy to prove:
a(n) = Sum_{d|n} A003602(d) * A026741(n/d).
a(n) = Sum_{d|n} A109168(d) * A193356(n/d), where A109168(d) = A140472(d) = (d+A006519(d))/2.
(End)
MATHEMATICA
nmax = 65; CoefficientList[Series[Sum[k x^k/(1 - x^(2 k))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, (n Mod[#, 2] + Boole[OddQ[n/#]] #)/2 &]; Table[a[n], {n, 1, 65}]
PROG
(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
(PARI) A328203(n) = if(n%2, (1/2)*(sigma(n)+(n*numdiv(n))), 2*A328203(n/2)); \\ Antti Karttunen, Nov 13 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Oct 07 2019
STATUS
approved