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A301746 Expansion of Product_{k>=1} (1 + x^k)^(sigma_0(k)^2). 2

%I #9 Aug 29 2018 17:55:14

%S 1,1,4,8,19,35,82,142,291,524,989,1724,3174,5393,9517,16064,27464,

%T 45481,76357,124402,204497,329559,532316,846564,1349481,2120814,

%U 3335819,5191522,8070062,12434176,19136484,29215324,44531151,67431985,101882975,153055897

%N Expansion of Product_{k>=1} (1 + x^k)^(sigma_0(k)^2).

%H Vaclav Kotesovec, <a href="/A301746/b301746.txt">Table of n, a(n) for n = 0..10000</a>

%F Conjecture: log(a(n)) ~ sqrt(n) * log(n)^(3/2) / (2*sqrt(6)). - _Vaclav Kotesovec_, Aug 29 2018

%t nmax = 50; CoefficientList[Series[Product[(1+x^k)^(DivisorSigma[0, k]^2), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k]^2, j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]]; , {k, 2, nmax}]; CoefficientList[s, x] (* _Vaclav Kotesovec_, Aug 29 2018 *)

%Y Cf. A000005, A035116, A107742, A301747.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Mar 26 2018

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