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A318917 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k/k), where phi is the Euler totient function A000010.

%I A318917 #26 Feb 28 2024 10:49:00
%S A318917 1,1,2,8,38,262,1732,16144,153596,1660796,19415384,264084064,
%T A318917 3664187848,57366995272,936097392752,16131362629568,302946516251408,
%U A318917 6034409270818576,125044362929875744,2756094464546395264,63280996793936902496
%N A318917 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k/k), where phi is the Euler totient function A000010.
%H A318917 Vaclav Kotesovec, <a href="/A318917/b318917.txt">Table of n, a(n) for n = 0..440</a>
%F A318917 a(n)/n! ~ 3^(1/4) * exp(2*sqrt(6*n)/Pi) / (Pi * 2^(3/4) * n^(3/4)).
%F A318917 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} mu(gcd(k,j)). - _Ilya Gutkovskiy_, Aug 17 2021
%F A318917 a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} phi(k) * a(n-k)/(n-k)!. - _Seiichi Manyama_, Apr 29 2022
%F A318917 E.g.f.: exp( Sum_{n>=1} (mu(n)/n) * x^n/(1 - x^n) ), where mu(n) = A008683(n). - _Paul D. Hanna_, Jun 24 2023
%t A318917 nmax = 20; CoefficientList[Series[Exp[Sum[EulerPhi[k]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!
%t A318917 a[n_] := a[n] = If[n == 0, 1, Sum[EulerPhi[k]* a[n-k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 20}]
%o A318917 (PARI) a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, eulerphi(k)*a(n-k)/(n-k)!)); \\ _Seiichi Manyama_, Apr 29 2022
%Y A318917 Cf. A088009, A318769, A318696.
%Y A318917 Cf. A000262, A305127, A318695.
%Y A318917 Cf. A053529, A028342.
%Y A318917 Cf. A000010 (phi), A008683 (mu).
%K A318917 nonn
%O A318917 0,3
%A A318917 _Vaclav Kotesovec_, Sep 05 2018