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A308564 Expansion of e.g.f. Sum_{k>=1} phi(k)*(exp(x) - 1)^k/k!, where phi = Euler totient function (A000010).

%I A308564 #10 Aug 04 2021 09:36:33
%S A308564 1,2,6,22,90,404,1974,10366,57864,341690,2134022,14104624,98498972,
%T A308564 723664482,5561589508,44473028634,368602225688,3159852790392,
%U A308564 27997141025686,256410638073082,2428063270357748,23774001479212114,240580239864321604,2513553050765310236
%N A308564 Expansion of e.g.f. Sum_{k>=1} phi(k)*(exp(x) - 1)^k/k!, where phi = Euler totient function (A000010).
%C A308564 Stirling transform of A000010.
%H A308564 Alois P. Heinz, <a href="/A308564/b308564.txt">Table of n, a(n) for n = 1..575</a>
%F A308564 G.f.: Sum_{k>=1} phi(k)*x^k / Product_{j=1..k} (1 - j*x).
%F A308564 a(n) = Sum_{k=1..n} Stirling2(n,k)*phi(k).
%p A308564 b:= proc(n, m) option remember; uses numtheory;
%p A308564      `if`(n=0, phi(m), m*b(n-1, m)+b(n-1, m+1))
%p A308564     end:
%p A308564 a:= n-> b(n, 0):
%p A308564 seq(a(n), n=1..24);  # _Alois P. Heinz_, Aug 04 2021
%t A308564 nmax = 24; Rest[CoefficientList[Series[Sum[EulerPhi[k] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
%t A308564 nmax = 24; Rest[CoefficientList[Series[Sum[EulerPhi[k] x^k/Product[(1 - j x), {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x]]
%t A308564 Table[Sum[StirlingS2[n, k] EulerPhi[k], {k, 1, n}], {n, 1, 24}]
%Y A308564 Cf. A000010, A306988.
%K A308564 nonn
%O A308564 1,2
%A A308564 _Ilya Gutkovskiy_, Jun 07 2019