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Expansion of Product_{k>=1} 1/(1 - k*x^k)^sigma(k), where sigma = A000203.
2

%I #11 Aug 27 2018 10:04:22

%S 1,1,7,19,71,173,583,1443,4255,10648,28929,71159,184740,445626,

%T 1110122,2638328,6369490,14870194,35031627,80465028,185556696,

%U 419916149,950785580,2121471778,4727971847,10412230698,22876886529,49776871862,107974178843,232302695301

%N Expansion of Product_{k>=1} 1/(1 - k*x^k)^sigma(k), where sigma = A000203.

%H Vaclav Kotesovec, <a href="/A318483/b318483.txt">Table of n, a(n) for n = 0..5920</a>

%F a(n) ~ c * n^3 * 3^(n/3), where

%F c = 280631952508395331283883354935233682635.581151020... if mod(n,3)=0

%F c = 280631952508395331283883354935233682635.059082354... if mod(n,3)=1

%F c = 280631952508395331283883354935233682635.088610121... if mod(n,3)=2

%F In closed form, c = (Product_{k>=4}((1 - k/3^(k/3))^(-sigma(k)))/(18*(57 - 90*3^(1/3) + 35*3^(2/3)))) - Product_{k>=4}((1 + ((-1)^(1 + 2*k/3)*k)/3^(k/3))^(-sigma(k)))/ ((-1)^(2*n/3)*(6*(3 + 2*(-3)^(1/3))^3*(-3 + (-3)^(2/3)))) - ((-1)^(1 - (4*n)/3)*Product_{k>=4}((1 + ((-1)^(1 + 4*k/3)*k)/3^(k/3))^(-sigma(k))))/(486*(1 + (-1/3)^(1/3))* (1 - 2*(-1/3)^(2/3))^3)

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

%t nmax = 40; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[1, k], j]*(-1)^j*k^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]

%Y Cf. A000203, A006906, A266941, A318415, A006171, A061256, A318484.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Aug 27 2018