A307679 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^(1/k).

1, 1, 5, 35, 323, 3679, 49819, 781465, 13923545, 277563617, 6118251461, 147715469131, 3875706370315, 109781717161375, 3338229675519803, 108443658227589329, 3747688533281296049, 137273241169036231105, 5311844045472206624005, 216505267421266611639667, 9270689769095765333645651

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

E.g.f.: exp(Sum_{k>=1} d(k)*x^k/(k*(1 - x)^k)), where d(k) is the number of divisors of k (A000005).

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A028342(k)*n!/k!.

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 35*x^3/3! + 323*x^4/4! + 3679*x^5/5! + 49819*x^6/6! + 781465*x^7/7! + 13923545*x^8/8! + ...
log(A(x)) = x + 4*x^2/2 + 11*x^3/3 + 27*x^4/4 + 62*x^5/5 + 137*x^6/6 + 296*x^7/7 + 630*x^8/8 + 1326*x^9/9 + ... + A160399(k)*x^k/k + ...

Mathematica

nmax = 20; CoefficientList[Series[Product[1/(1 - x^k/(1 - x)^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

Crossrefs

Cf. A000005, A028342, A103446, A160399, A218482, A307680, A320563.

Sequence in context: A260952 A349515 A362356 * A305306 A113342 A262248

Adjacent sequences: A307676 A307677 A307678 * A307680 A307681 A307682

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 21 2019

Status

approved