A318483 Expansion of Product_{k>=1} 1/(1 - k*x^k)^sigma(k), where sigma = A000203.
1, 1, 7, 19, 71, 173, 583, 1443, 4255, 10648, 28929, 71159, 184740, 445626, 1110122, 2638328, 6369490, 14870194, 35031627, 80465028, 185556696, 419916149, 950785580, 2121471778, 4727971847, 10412230698, 22876886529, 49776871862, 107974178843, 232302695301
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..5920
Programs
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Mathematica
nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^DivisorSigma[1, k], {k, 1, nmax}], {x, 0, nmax}], x] 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]
Formula
a(n) ~ c * n^3 * 3^(n/3), where
c = 280631952508395331283883354935233682635.581151020... if mod(n,3)=0
c = 280631952508395331283883354935233682635.059082354... if mod(n,3)=1
c = 280631952508395331283883354935233682635.088610121... if mod(n,3)=2
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)