A318912 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(mu(k)^2/k), where mu = Möbius function (A008683).

1, 1, 3, 11, 53, 309, 2359, 18367, 168489, 1690217, 19416491, 233144691, 3187062493, 44901291421, 700058510943, 11509417045799, 200586478516049, 3680237286827217, 72326917665944659, 1467930587827522267, 31855597406715020421, 718484783876745110021, 16993553696264436052103

Offset

0,3

Links

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

Formula

E.g.f.: exp(Sum_{k>=1} 2^omega(k)*x^k/k), where omega(k) = number of distinct primes dividing k (A001221).

Maple

seq(n!*coeff(series(mul(1/(1-x^k)^(mobius(k)^2/k), k=1..100), x=0, 23), x, n), n=0..22); # Paolo P. Lava, Jan 09 2019

Mathematica

nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(MoebiusMu[k]^2/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[2^PrimeNu[k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[2^PrimeNu[k] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

Crossrefs

Cf. A001221, A008683, A028342, A034444, A073576, A206303, A318913, A318914.

Sequence in context: A305710 A351200 A000255 * A121580 A321732 A224345

Adjacent sequences: A318909 A318910 A318911 * A318913 A318914 A318915

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 05 2018

Status

approved