A301554 - OEIS

%I #26 Sep 08 2022 08:46:20

%S 1,2,6,14,32,66,138,266,512,948,1730,3074,5408,9306,15854,26594,44150,

%T 72378,117620,189074,301516,476518,747514,1163470,1798920,2762040,

%U 4215194,6393196,9642596,14462518,21581386,32040562,47345342,69635866,101974722,148692638

%N Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(sigma_0(k)).

%C Convolution of A006171 and A107742.

%H Vaclav Kotesovec, <a href="/A301554/b301554.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))/(1 - x^(i*j)). - _Ilya Gutkovskiy_, May 23 2018

%F Conjecture: log(a(n)) ~ Pi * sqrt(n*log(n)/2). - _Vaclav Kotesovec_, Sep 03 2018

%p with(numtheory): seq(coeff(series(mul(((1+x^k)/(1-x^k))^sigma[0](k),k=1..n),x,n+1), x, n), n = 0 .. 35); # _Muniru A Asiru_, Oct 29 2018

%t nmax = 50; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]

%o (PARI) m=50; x='x+O('x^m); Vec(prod(k=1,m, prod(j=1,m+2, (1+x^(j*k))/(1-x^(j*k)) ))) \\ _G. C. Greubel_, Oct 29 2018

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[(&*[(1 + x^(j*k))/(1-x^(j*k)): j in [1..(m+2)]]): k in [1..(m+2)]]))); // _G. C. Greubel_, Oct 29 2018

%Y Cf. A000005, A006171, A107742, A320237.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Mar 23 2018