A318693 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(i*j)/(i*j)).

1, 1, 4, 16, 100, 628, 5388, 46212, 491328, 5381760, 68023056, 892073136, 13238778144, 201822014496, 3397195558560, 59356290115296, 1121097742183296, 21916440531679104, 459855848691876096, 9952944631606759680, 229191463614349301760, 5446997871156332605440, 136439919208493792455680

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

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

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

Maple

seq(n!*coeff(series(mul(mul(1/(1-x^(i*j)/(i*j)), i=1..30), j=1..30), x=0, 23), x, n), n=0..22); # Paolo P. Lava, Jan 09 2019

Mathematica

nmax = 22; CoefficientList[Series[Product[Product[1/(1 - x^(i j)/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[1/(1 - x^k/k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[d^(1 - k/d) DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^(1 - k/d) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

Crossrefs

Cf. A000005, A006171, A007841, A294469, A318415, A318694.

Sequence in context: A114023 A145823 A121142 * A087296 A065731 A074187

Adjacent sequences: A318690 A318691 A318692 * A318694 A318695 A318696

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 31 2018

Status

approved