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

%I A305305 #8 Jun 09 2018 01:54:33
%S A305305 1,1,5,32,292,3174,42758,659028,11725656,233646240,5183599152,
%T A305305 126353158656,3362529785712,96896454983184,3007687250735568,
%U A305305 100017757744279584,3547903924884082176,133715849506895518848,5336112511923188151168,224772952826373341478912,9966476790792153522756864
%N A305305 Expansion of e.g.f. 1/(1 - Sum_{k>=1} x^k/(k*(1 - x^k))).
%C A305305 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, ...].
%H A305305 Alois P. Heinz, <a href="/A305305/b305305.txt">Table of n, a(n) for n = 0..397</a>
%H A305305 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A305305 E.g.f.: 1/(1 - Sum_{k>=1} (sigma(k)/k)*x^k), where sigma() = A000203.
%F A305305 E.g.f.: 1/(1 - Sum_{k>=1} (A017665(k)/A017666(k))*x^k).
%F A305305 E.g.f.: 1/(1 - log(f(x))), where f(x) = o.g.f. for A000041, Product_{k>=1} 1/(1 - x^k).
%e A305305 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! + ...
%p A305305 b:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A305305       1/d, d=numtheory[divisors](j))*b(n-j), j=1..n))
%p A305305     end:
%p A305305 a:= n-> b(n)*n!:
%p A305305 seq(a(n), n=0..20);  # _Alois P. Heinz_, May 29 2018
%t A305305 nmax = 20; CoefficientList[Series[1/(1 - Sum[x^k/(k (1 - x^k)), {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
%t A305305 nmax = 20; CoefficientList[Series[1/(1 - Sum[DivisorSigma[-1, k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
%t A305305 a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[-1, k] a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 20}]
%Y A305305 Cf. A000041, A000203, A017665, A017666, A038048, A180305.
%K A305305 nonn
%O A305305 0,3
%A A305305 _Ilya Gutkovskiy_, May 29 2018