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Expansion of e.g.f. 1/(1 - Sum_{k>=1} x^k/(k*(1 - x^k))).
2

%I #8 Jun 09 2018 01:54:33

%S 1,1,5,32,292,3174,42758,659028,11725656,233646240,5183599152,

%T 126353158656,3362529785712,96896454983184,3007687250735568,

%U 100017757744279584,3547903924884082176,133715849506895518848,5336112511923188151168,224772952826373341478912,9966476790792153522756864

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

%C 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 Alois P. Heinz, <a href="/A305305/b305305.txt">Table of n, a(n) for n = 0..397</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

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

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

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

%e 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 b:= proc(n) option remember; `if`(n=0, 1, add(add(

%p 1/d, d=numtheory[divisors](j))*b(n-j), j=1..n))

%p end:

%p a:= n-> b(n)*n!:

%p seq(a(n), n=0..20); # _Alois P. Heinz_, May 29 2018

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

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

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 29 2018