%I #45 Oct 28 2018 03:47:23

%S 1,1,6,16,52,128,373,913,2399,5796,14298,33655,79756,183078,419846,

%T 942807,2106176,4633208,10127557,21870997,46912648,99639685,210206722,

%U 439777198,914157490,1886428608,3869204040,7884691072,15976273573,32182538964,64484592372,128518359868,254868985099,502950483815,987904826874,1931596634076

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

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

%H Seiichi Manyama, <a href="/A275585/b275585.txt">Table of n, a(n) for n = 0..10000</a>

%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>

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

%F a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A027847(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Jun 08 2017

%F 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

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

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(

%p d*sigma[2](d), d=divisors(j))*a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..40); # _Alois P. Heinz_, Jun 08 2017

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

%Y Cf. A001157, A027847, A288414.

%Y 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).

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Dec 25 2016