A328203 Expansion of Sum_{k>=1} k * x^k / (1 - x^(2*k))^2.

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

Antti Karttunen, Table of n, a(n) for n = 1..20000

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

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

Cf. also A347957.

Sequence in context: A243973 A286015 A183542 * A080031 A198193 A316905

Adjacent sequences: A328200 A328201 A328202 * A328204 A328205 A328206

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 07 2019

Status

approved