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A326830
Expansion of Product_{i>=2, j>=2} 1 / (1 - x^(i*j))^j.
2
1, 0, 0, 0, 2, 0, 5, 0, 9, 3, 17, 0, 46, 6, 68, 23, 153, 27, 297, 67, 534, 188, 978, 276, 1932, 620, 3250, 1313, 6033, 2246, 10854, 4361, 18776, 8639, 32831, 14835, 58230, 27635, 98052, 50980, 169522, 88243, 289720, 157179, 486232, 280206, 818006, 478014
OFFSET
0,5
COMMENTS
Euler transform of A048050.
Convolution of A326830 and A002865 is A318784. - Vaclav Kotesovec, Oct 26 2019
LINKS
FORMULA
G.f.: Product_{k>=1} 1 / (1 - x^k)^A048050(k).
G.f.: exp(Sum_{k>=1} (A001001(k) - A000203(k) - A001157(k) + 1) * x^k / k).
a(n) ~ exp(3^(2/3) * ((Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3)/2 - Pi^2 * (3/((Pi^2 - 6)*Zeta(3)))^(1/3) * n^(1/3)/4 - Pi^4 / (32*(Pi^2 - 6)*Zeta(3)) - 1/8) * A^(3/2)* (2*Pi)^(1/24) / (3^(1/8) * ((Pi^2 - 6)*Zeta(3))^(3/8) * n^(1/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 26 2019
MAPLE
with(numtheory):
b:= proc(n) option remember; `if`(n<4, 0, sigma(n)-1-n) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..50); # Alois P. Heinz, Oct 20 2019
MATHEMATICA
nmax = 47; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k] - k - 1), {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[If[d == 1, 0, d (DivisorSigma[1, d] - d - 1)], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 47}]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Oct 20 2019
STATUS
approved