A301747 Expansion of Product_{k>=1} (1/(1 - x^k))^(sigma_0(k)^2).
1, 1, 5, 9, 28, 48, 130, 226, 532, 941, 2021, 3545, 7210, 12509, 24209, 41715, 77742, 132404, 239655, 403731, 712426, 1188079, 2052070, 3386854, 5745200, 9388740, 15672560, 25376167, 41765597, 67021171, 108932532, 173327693, 278533669, 439653317, 699265665
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Programs
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Mathematica
nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^(DivisorSigma[0, k]^2), {k, 1, nmax}], {x, 0, nmax}], x] nmax = 50; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[0, k]^2, j]*(-1)^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[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 29 2018 *)
Formula
log(a(n)) ~ sqrt(n) * log(n)^(3/2) / (2*sqrt(3)). - Vaclav Kotesovec, Aug 28 2018