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A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.

%I A318811 #27 Feb 28 2024 10:48:53
%S A318811 1,1,3,19,121,1161,9931,124363,1542129,21594961,335083411,5712781251,
%T A318811 104044684393,2036445474649,42781075481691,943820382272251,
%U A318811 22433542236603361,556276331238284193,14612462927067954979,401110580118493111411,11553483337639043003481
%N A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.
%H A318811 Seiichi Manyama, <a href="/A318811/b318811.txt">Table of n, a(n) for n = 0..435</a>
%H A318811 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotientFunction.html">Totient Function</a>.
%H A318811 Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler%27s_totient_function">Euler's totient function</a>.
%F A318811 a(n) ~ 2^(1/3) * exp(1/6 + 3^(4/3) * n^(2/3) / (2^(1/3) * Pi^(2/3)) - n) * n^(n - 1/6) / (3*Pi)^(1/3).
%F A318811 a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * phi(k) * a(n-k)/(n-k)!. - _Seiichi Manyama_, Apr 07 2022
%t A318811 nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!
%o A318811 (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, eulerphi(k)*x^k)))) \\ _Seiichi Manyama_, Apr 07 2022
%o A318811 (PARI) a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*eulerphi(k)*a(n-k)/(n-k)!)); \\ _Seiichi Manyama_, Apr 07 2022
%Y A318811 Cf. A061255, A061256, A006171.
%Y A318811 Cf. A299069, A192065, A107742.
%Y A318811 Cf. A294361, A294363.
%K A318811 nonn
%O A318811 0,3
%A A318811 _Vaclav Kotesovec_, Sep 04 2018