A318914 Expansion of e.g.f. Product_{p prime, k>=1} 1/(1 - x^(p^k))^(1/(p^k)).

1, 0, 1, 2, 15, 44, 475, 2274, 33313, 227240, 2920041, 26754650, 469513231, 4613913732, 85842524755, 1174844041994, 24672317426625, 334246510927184, 7985602649948113, 127351500133158450, 3282809137540001551, 60776696924693716700, 1556379682561575238731, 32568139442090869594802

Offset: 0

Author

Ilya Gutkovskiy, Sep 05 2018

Keywords

• nonn

Crossrefs

Cf. A001222, A023894, A028342, A206303, A318912, A318913.

Programs

Maple

seq(n!*coeff(series(exp(add(bigomega(k)*x^k/k,k=1..100)),x=0,24),x,n),n=0..23); # Paolo P. Lava, Jan 09 2019

Mathematica

nmax = 23; CoefficientList[Series[Product[1/(1 - x^k)^(Boole[PrimePowerQ[k]]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[Sum[PrimeOmega[k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[PrimeOmega[k] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

Formula

E.g.f.: exp(Sum_{k>=1} bigomega(k)*x^k/k), where bigomega(k) = number of prime divisors of k counted with multiplicity (A001222).