A319130 Expansion of Product_{k>=1} 1/(1 - x^k)^(2^omega(k)), where omega(k) = number of distinct primes dividing k (A001221).

1, 1, 3, 5, 10, 16, 31, 47, 81, 125, 203, 305, 482, 710, 1082, 1582, 2348, 3380, 4933, 7007, 10048, 14136, 19972, 27796, 38822, 53510, 73903, 101033, 138165, 187351, 254055, 341923, 459956, 614904, 821162, 1090740, 1447109, 1910665, 2519325, 3308019, 4336956

Offset: 0

Author

Ilya Gutkovskiy, Sep 11 2018

Keywords

• nonn

Comments

Euler transform of A034444.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A001221, A006171, A008683, A034444, A048250, A293548.

Programs

Maple

with(numtheory): a:=series(mul(1/(1-x^k)^(2^nops(factorset(k))),k=1..50),x=0,41): seq(coeff(a,x,n),n=0..40); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 40; CoefficientList[Series[Product[1/(1 - x^k)^(2^PrimeNu[k]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[DivisorSum[k, # &, SquareFreeQ[#] &] x^k/(k (1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d 2^PrimeNu[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A034444(k).

G.f.: exp(Sum_{k>=1} A048250(k)*x^k/(k*(1 - x^k))).

G.f.: exp(Sum_{k>=1} Sum_{j>=1} mu(j)^2*x^(j*k)/(k*(1 - x^(j*k)))), where mu = Möbius function (A008683).

log(a(n)) ~ sqrt(2*n*log(n)). - Vaclav Kotesovec, Sep 13 2018