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A288414 Expansion of Product_{k>=1} (1 + x^k)^(sigma_2(k)).

%I A288414 #30 Sep 08 2022 08:46:19
%S A288414 1,1,5,15,41,107,286,700,1735,4162,9803,22673,51822,116376,258548,
%T A288414 567197,1230763,2642958,5622616,11850537,24769248,51353095,105662389,
%U A288414 215838649,437890022,882562763,1767741732,3519599996,6967592060,13717874719,26865949075
%N A288414 Expansion of Product_{k>=1} (1 + x^k)^(sigma_2(k)).
%H A288414 Seiichi Manyama, <a href="/A288414/b288414.txt">Table of n, a(n) for n = 0..10000</a>
%F A288414 a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288419(k)*a(n-k) for n > 0.
%F A288414 a(n) ~ exp(2^(5/4) * (7*Zeta(3))^(1/4) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - 5^(1/4) * Pi * n^(1/4) / (2^(13/4) * 3^(7/4) * (7*Zeta(3))^(1/4))) * (7*Zeta(3))^(1/8) / (2^(15/8) * 15^(1/8) * n^(5/8)). - _Vaclav Kotesovec_, Mar 23 2018
%F A288414 G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))^(j^2). - _Ilya Gutkovskiy_, Aug 26 2018
%p A288414 with(numtheory): seq(coeff(series(mul((1+x^k)^(sigma[2](k)),k=1..n),x,n+1), x, n), n = 0 .. 30); # _Muniru A Asiru_, Oct 31 2018
%t A288414 nmax = 40; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[2, k], {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 09 2017 *)
%o A288414 (PARI) m=40; x='x+O('x^m); Vec(prod(k=1, m, (1+x^k)^sigma(k,2))) \\ _G. C. Greubel_, Oct 30 2018
%o A288414 (Magma) m:=40; R<q>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+q^k)^DivisorSigma(2,k): k in [1..m]]) )); // _G. C. Greubel_, Oct 30 2018
%Y A288414 Cf. A275585, A288389, A288419.
%Y A288414 Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), A192065 (m=1), this sequence (m=2), A288415 (m=3).
%K A288414 nonn
%O A288414 0,3
%A A288414 _Seiichi Manyama_, Jun 08 2017