A318415 Expansion of Product_{i>=1, j>=1} 1/(1 - i*j*x^(i*j)).

1, 1, 5, 11, 35, 69, 200, 398, 1014, 2069, 4820, 9716, 21787, 43209, 92530, 182773, 378676, 737526, 1492451, 2872788, 5686194, 10837935, 21052463, 39699970, 75972300, 141818166, 267607065, 495142606, 922920753, 1692529453, 3121105278, 5676677651, 10364752129, 18708292447, 33851433117, 60656841965

Offset: 0

Author

Ilya Gutkovskiy, Aug 26 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000005, A006171, A006906, A280540, A318416.

Programs

Mathematica

nmax = 35; CoefficientList[Series[Product[Product[1/(1 - i j x^(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[Product[1/(1 - k x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[Exp[Sum[Sum[d^(k/d + 1) DivisorSigma[0, 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^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 35}]
nmax = 40; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[0, 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] (* Vaclav Kotesovec, Aug 27 2018 *)

Formula

G.f.: Product_{k>=1} 1/(1 - k*x^k)^tau(k), where tau = number of divisors (A000005).

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d^(k/d+1)*tau(d) ) * x^k/k).

From Vaclav Kotesovec, Aug 27 2018: (Start)

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

c = 10751825728554.298582954430359167227238488440778317... if mod(n,3)=0

c = 10751825728553.835664124121831524829543267756895348... if mod(n,3)=1

c = 10751825728553.838520991588115910603754564083195806... if mod(n,3)=2

In closed form, c = (Product_{k>=4}((1 - k/3^(k/3))^(-sigma(0,k)))) / (21 - 16*3^(1/3) + 3^(2/3)) - (3*Product_{k>=4}((1 + ((-1)^(1 + 2*k/3)*k)/3^(k/3))^(-sigma(0,k)))) / ((-1)^(2*n/3)*((3 + 2*(-3)^(1/3))^2*(-3 + (-3)^(2/3)))) + Product_{k>=4}((1 + ((-1)^(1 + 4*k/3)*k)/3^(k/3))^(-sigma(0,k))) / (9*(-1)^(4*n/3)*((1 + (-1/3)^(1/3))*(1 - 2*(-1/3)^(2/3))^2))

(End)