A275585 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_2(k)).

1, 1, 6, 16, 52, 128, 373, 913, 2399, 5796, 14298, 33655, 79756, 183078, 419846, 942807, 2106176, 4633208, 10127557, 21870997, 46912648, 99639685, 210206722, 439777198, 914157490, 1886428608, 3869204040, 7884691072, 15976273573, 32182538964, 64484592372, 128518359868, 254868985099, 502950483815, 987904826874, 1931596634076

Offset

0,3

Comments

Euler transform of the sum of squares of divisors (A001157).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

N. J. A. Sloane, Transforms

Index entries for sequences related to sums of divisors

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^(sigma_2(k)).

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A027847(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 08 2017

a(n) ~ exp(4*Pi * Zeta(3)^(1/4) * n^(3/4) / (3^(5/4) * 5^(1/4)) - Pi * 5^(1/4) * n^(1/4) / (8 * 3^(7/4) * Zeta(3)^(1/4)) + Zeta(3) / (8*Pi^2)) * Zeta(3)^(1/8) / (2^(3/2) * 15^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 23 2018

G.f.: exp(Sum_{k>=1} sigma_3(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma[2](d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 08 2017

Mathematica

nmax = 35; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[2, k]), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A001157, A027847, A288414.

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), this sequence (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

Sequence in context: A288572 A367993 A301978 * A026086 A175659 A221270

Adjacent sequences: A275582 A275583 A275584 * A275586 A275587 A275588

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 25 2016

Status

approved