|
%I #19 Mar 10 2023 10:13:56
%S 1,1,7,16,61,130,429,945,2684,5990,15530,34313,83995,183070,427046,
%T 919480,2067589,4384678,9577536,20019243,42664087,87954522,183573639,
%U 373430131,765524808,1537737243,3102614407,6159028445,12252086879,24051526041,47239506797,91765428710,178156003047
%N Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - x^(i*j*k))^(i*j*k).
%H Vaclav Kotesovec, <a href="/A318413/b318413.txt">Table of n, a(n) for n = 0..10000</a>
%H Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, <a href="https://arxiv.org/abs/2303.02240">A unified treatment of families of partition functions</a>, arXiv:2303.02240 [math.CO], 2023.
%F G.f.: Product_{k>=1} 1/(1 - x^k)^(k*tau_3(k)), where tau_3() = A007425.
%F G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d^2 * Sum_{j|d} tau(j) ) * x^k/k), where tau() = A000005.
%F Conjecture: log(a(n)) ~ (3*Zeta(3))^(1/3) * log(n)^(2/3) * n^(2/3) / 2. - _Vaclav Kotesovec_, Sep 02 2018
%p a:=series(mul(mul(mul(1/(1-x^(i*j*k))^(i*j*k),k=1..55),j=1..55),i=1..55),x=0,33): seq(coeff(a,x,n),n=0..32); # _Paolo P. Lava_, Apr 02 2019
%t nmax = 32; CoefficientList[Series[Product[Product[Product[1/(1 - x^(i j k))^(i j k), {i, 1, nmax}], {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 32; CoefficientList[Series[Product[1/(1 - x^k)^(k Sum[DivisorSigma[0, d], {d, Divisors[k]}]), {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 32; CoefficientList[Series[Exp[Sum[Sum[d^2 Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]
%t nmax = 50; A034718 = Table[n*Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, nmax}]; s = 1 - x; Do[s *= Sum[Binomial[A034718[[k]], 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 31 2018 *)
%Y Cf. A000005, A007425, A034718, A174465, A174467, A280540, A318414.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 26 2018
|
