A288391 - OEIS

%I #27 Sep 08 2022 08:46:19

%S 1,1,10,38,156,534,2014,6796,23312,76165,247234,780343,2435903,

%T 7453859,22538336,67130594,197666509,574876417,1654464954,4711217687,

%U 13288453688,37133349758,102873771662,282630567325,770410193747,2084205092693,5598070811010

%N Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_3(k)).

%H Seiichi Manyama, <a href="/A288391/b288391.txt">Table of n, a(n) for n = 0..1000</a>

%F a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A027848(k)*a(n-k) for n > 0.

%F a(n) ~ exp((5*Pi)^(4/5) * Zeta(5)^(1/5) * n^(4/5) / (2^(8/5) * 3^(1/5)) - Zeta'(-3)/2) * Zeta(5)^(121/1200) / ((24*Pi)^(121/1200) * 5^(721/1200) * n^(721/1200)). - _Vaclav Kotesovec_, Mar 23 2018

%F G.f.: exp(Sum_{k>=1} sigma_4(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[3](d), d=divisors(j))*a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Jun 08 2017

%t nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 23 2018 *)

%o (PARI) m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^sigma(k,3))) \\ _G. C. Greubel_, Oct 30 2018

%o (Magma) m:=40; R<q>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-q^k)^DivisorSigma(3,k): k in [1..m]]) )); // _G. C. Greubel_, Oct 30 2018

%Y Cf. A027848, A288392, A288415.

%Y Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), A275585 (m=2), this sequence (m=3).

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jun 08 2017