A318413 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - x^(i*j*k))^(i*j*k).
1, 1, 7, 16, 61, 130, 429, 945, 2684, 5990, 15530, 34313, 83995, 183070, 427046, 919480, 2067589, 4384678, 9577536, 20019243, 42664087, 87954522, 183573639, 373430131, 765524808, 1537737243, 3102614407, 6159028445, 12252086879, 24051526041, 47239506797, 91765428710, 178156003047
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
- Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
Programs
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Maple
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
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Mathematica
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] nmax = 32; CoefficientList[Series[Product[1/(1 - x^k)^(k Sum[DivisorSigma[0, d], {d, Divisors[k]}]), {k, 1, nmax}], {x, 0, nmax}], x] 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] 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}] 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 *)
Formula
G.f.: Product_{k>=1} 1/(1 - x^k)^(k*tau_3(k)), where tau_3() = A007425.
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d^2 * Sum_{j|d} tau(j) ) * x^k/k), where tau() = A000005.
Conjecture: log(a(n)) ~ (3*Zeta(3))^(1/3) * log(n)^(2/3) * n^(2/3) / 2. - Vaclav Kotesovec, Sep 02 2018
Comments