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%I #31 Sep 08 2022 08:46:19
%S 1,1,9,37,137,487,1749,5901,19695,63832,202905,632689,1941394,5860868,
%T 17448558,51255292,148726841,426605755,1210569740,3400427281,
%U 9460683203,26083933370,71300381025,193313191005,520057831035,1388722752205,3682100198763
%N Expansion of Product_{k>=1} (1 + x^k)^(sigma_3(k)).
%H Seiichi Manyama, <a href="/A288415/b288415.txt">Table of n, a(n) for n = 0..5402</a>
%F a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288420(k)*a(n-k) for n > 0.
%F a(n) ~ exp(5*Pi^(4/5) * Zeta(5)^(1/5) * n^(4/5) / 2^(12/5)) * Zeta(5)^(1/10) / (2^(169/240) * sqrt(5) * Pi^(1/10) * n^(3/5)). - _Vaclav Kotesovec_, Mar 23 2018
%F G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))^(j^3). - _Ilya Gutkovskiy_, Aug 26 2018
%p with(numtheory): seq(coeff(series(mul((1+x^k)^(sigma[3](k)),k=1..n),x,n+1), x, n), n = 0 .. 30); # _Muniru A Asiru_, Oct 31 2018
%t nmax = 40; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 09 2017 *)
%o (PARI) m=40; x='x+O('x^m); Vec(prod(k=1, m, (1+x^k)^sigma(k,3))) \\ _G. C. Greubel_, Oct 30 2018
%o (Magma) m:=40; R<q>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+q^k)^DivisorSigma(3,k): k in [1..m]]) )); // _G. C. Greubel_, Oct 30 2018
%Y Cf. A288391, A288392, A288420.
%Y Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), A192065 (m=1), A288414 (m=2), this sequence (m=3).
%K nonn
%O 0,3
%A _Seiichi Manyama_, Jun 09 2017