A305305 Expansion of e.g.f. 1/(1 - Sum_{k>=1} x^k/(k*(1 - x^k))).

1, 1, 5, 32, 292, 3174, 42758, 659028, 11725656, 233646240, 5183599152, 126353158656, 3362529785712, 96896454983184, 3007687250735568, 100017757744279584, 3547903924884082176, 133715849506895518848, 5336112511923188151168, 224772952826373341478912, 9966476790792153522756864

Offset: 0

Author

Ilya Gutkovskiy, May 29 2018

Keywords

• nonn

Comments

a(n)/n! is the invert transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 32*x^3/3! + 292*x^4/4! + 3174*x^5/5! + 42758*x^6/6! + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..397
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000041, A000203, A017665, A017666, A038048, A180305.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(add(
      1/d, d=numtheory[divisors](j))*b(n-j), j=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..20);  # Alois P. Heinz, May 29 2018

Mathematica

nmax = 20; CoefficientList[Series[1/(1 - Sum[x^k/(k (1 - x^k)), {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[1/(1 - Sum[DivisorSigma[-1, k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[-1, k] a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 20}]

Formula

E.g.f.: 1/(1 - Sum_{k>=1} (sigma(k)/k)*x^k), where sigma() = A000203.

E.g.f.: 1/(1 - Sum_{k>=1} (A017665(k)/A017666(k))*x^k).

E.g.f.: 1/(1 - log(f(x))), where f(x) = o.g.f. for A000041, Product_{k>=1} 1/(1 - x^k).