A113415 Expansion of Sum_{k>0} x^k/(1-x^(2k))^2.

1, 1, 3, 1, 4, 3, 5, 1, 8, 4, 7, 3, 8, 5, 14, 1, 10, 8, 11, 4, 18, 7, 13, 3, 17, 8, 22, 5, 16, 14, 17, 1, 26, 10, 26, 8, 20, 11, 30, 4, 22, 18, 23, 7, 42, 13, 25, 3, 30, 17, 38, 8, 28, 22, 38, 5, 42, 16, 31, 14, 32, 17, 55, 1, 44, 26, 35, 10, 50, 26, 37, 8, 38, 20, 65, 11, 50, 30, 41

Offset

1,3

Comments

Arithmetic mean between the number of odd divisors (A001227) and their sum (A000593). This fact was essentially found by the algorithmic search of Jon Maiga's Sequence Machine, and is easily seen to be correct when compared to the PARI-program given by the original author. - Antti Karttunen, Dec 07 2021

Links

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

Jon Maiga, Computer-generated formulas for A113415, Sequence Machine.

Formula

G.f.: Sum_{k>0} x^k/(1-x^(2k))^2 = Sum_{k>0} k x^(2k-1)/(1-x^(2k-1)).

a(n) = (1/2) * Sum_{d|n} (d+1)*(d mod 2). - Wesley Ivan Hurt, Nov 25 2021 [From PARI prog]

From Antti Karttunen, Dec 07 2021: (Start)

All these formulas, except the last, were found by the Sequence Machine in some form or another:

a(n) = (1/2) * (A000593(n)+A001227(n)).

a(n) = A069734(A000265(n)). [See either Rutherford's or Luschny's formula in A069734]

a(n) = A349371(n) / A001511(n).

a(n) = A349371(A000265(n)) = A336840(A064989(n)).

a(n) = a(2*n) = a(A000265(n)) = A349916(4*n).

(End)

Mathematica

Array[DivisorSum[#, If[OddQ[#], (# + 1)/2, 0] &] &, 79] (* Michael De Vlieger, Dec 08 2021 *)

Prog

(PARI) a(n)=if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)))

Crossrefs

Cf. A000265, A000593, A001227, A001511, A003602, A048673, A064989, A069734, A336840, A349371, A349915 (Dirichlet inverse).

Quadrisection of A349916.

Sequence in context: A335906 A029151 A102595 * A332801 A054019 A333477

Adjacent sequences: A113412 A113413 A113414 * A113416 A113417 A113418

Keyword

nonn

Author

Michael Somos, Oct 29 2005

Status

approved