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Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k/k!), where phi() is the Euler totient function (A000010).
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%I #22 Feb 27 2022 10:21:20

%S 1,1,2,6,20,80,362,1820,10084,60522,391864,2714514,20001700,156107224,

%T 1284705246,11112088358,100698613720,953478331288,9410963022318,

%U 96614921664444,1029705968813656,11373102766644372,129972789566984682,1534638410054873892,18696544357738885720

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

%C Exponential transform of A000010.

%H Alois P. Heinz, <a href="/A300011/b300011.txt">Table of n, a(n) for n = 0..552</a>

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

%F E.g.f.: exp(Sum_{k>=1} A000010(k)*x^k/k!).

%F a(0) = 1; a(n) = Sum_{k=1..n} phi(k) * binomial(n-1,k-1) * a(n-k). - _Seiichi Manyama_, Feb 27 2022

%e E.g.f.: A(x) = 1 + x/1! + 2*x^2/2! + 6*x^3/3! + 20*x^4/4! + 80*x^5/5! + 362*x^6/6! + 1820*x^7/7! + ...

%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*

%p binomial(n-1, j-1)*numtheory[phi](j), j=1..n))

%p end:

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Mar 09 2018

%t nmax = 24; CoefficientList[Series[Exp[Sum[EulerPhi[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

%t a[n_] := a[n] = Sum[EulerPhi[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}]

%o (PARI) a(n) = if(n==0, 1, sum(k=1, n, eulerphi(k)*binomial(n-1, k-1)*a(n-k))); \\ _Seiichi Manyama_, Feb 27 2022

%Y Cf. A000010, A050392, A159929, A274804, A295739.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Mar 09 2018