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A318414 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))^(i*j*k). 6

%I #20 Mar 10 2023 10:13:59

%S 1,1,6,15,48,108,323,716,1868,4217,10137,22311,51477,110817,245260,

%T 519918,1114914,2318557,4854952,9923533,20335761,40941170,82365742,

%U 163413699,323589060,633429923,1236392498,2390718266,4606489839,8805346615,16768968317,31713677061,59747953446

%N Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))^(i*j*k).

%H Vaclav Kotesovec, <a href="/A318414/b318414.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 + x^k)^(k*tau_3(k)), where tau_3() = A007425.

%F G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d^2 * Sum_{j|d} tau(j) ) * x^k/k), where tau() = A000005.

%F Conjecture: log(a(n)) ~ 3^(2/3) * Zeta(3)^(1/3) * log(n)^(2/3) * n^(2/3) / 2^(5/3). - _Vaclav Kotesovec_, Sep 02 2018

%p a:=series(mul(mul(mul((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 + 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 + 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[(-1)^(k/d + 1) 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[(-1)^(k/d + 1) 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 = 32; A034718 = Table[n*Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, nmax}]; s = 1 + x; Do[s *= Sum[Binomial[A034718[[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}]; Take[CoefficientList[s, x], nmax + 1] (* _Vaclav Kotesovec_, Aug 31 2018 *)

%Y Cf. A000005, A007425, A034718, A280473, A280541, A318413.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 26 2018

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Last modified March 28 11:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)