cp's OEIS Frontend

A328053 Expansion of e.g.f. log(1 + Sum_{k>=1} phi(k) * x^k / k!), where phi = Euler totient function (A000010).

%I A328053 #8 Oct 17 2019 05:43:16
%S A328053 0,1,0,1,-3,8,-32,166,-926,5842,-42812,348632,-3088388,29871372,
%T A328053 -314102574,3554714938,-43057252520,556487433400,-7644034688586,
%U A328053 111160926400032,-1706191876272876,27567942738717360,-467712309003533398,8312805777830133096
%N A328053 Expansion of e.g.f. log(1 + Sum_{k>=1} phi(k) * x^k / k!), where phi = Euler totient function (A000010).
%C A328053 Logarithmic transform of A000010.
%H A328053 Vaclav Kotesovec, <a href="/A328053/b328053.txt">Table of n, a(n) for n = 0..465</a>
%F A328053 a(n)/n! ~ -(-1)^n * d^n / n, where d = 0.8078801380543809482705136550646927880437760099284517780830096910529492372472... - _Vaclav Kotesovec_, Oct 17 2019
%p A328053 a:= proc(n) option remember; `if`(n=0, 0, (b-> b(n)-add(a(j)
%p A328053      *binomial(n, j)*j*b(n-j), j=1..n-1)/n)(numtheory[phi]))
%p A328053     end:
%p A328053 seq(a(n), n=0..25);  # _Alois P. Heinz_, Oct 06 2019
%t A328053 nmax = 23; CoefficientList[Series[Log[1 + Sum[EulerPhi[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A328053 a[n_] := a[n] = EulerPhi[n] - Sum[Binomial[n, k] EulerPhi[n - k] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 23}]
%Y A328053 Cf. A000010, A300011, A318811, A318917.
%K A328053 sign
%O A328053 0,5
%A A328053 _Ilya Gutkovskiy_, Oct 03 2019