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Expansion of e.g.f. 1/(1 - Sum_{k>=1} phi(k) * x^k / k), where phi is the Euler totient function A000010.
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%I #20 Apr 30 2022 12:19:12

%S 1,1,3,16,110,986,10202,126288,1770120,27939192,489658632,9455296896,

%T 198951693360,4537680805776,111426422418768,2931467216681856,

%U 82273083792879744,2453340521239749504,77458777017799833216,2581489882182061744128

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

%F a(0) = 1; a(n) = Sum_{k=1..n} A074930(k) * binomial(n,k) * a(n-k).

%t phi[k_] := phi[k] = EulerPhi[k]; a[0] = 1; a[n_] := a[n] = Sum[(k - 1)! * phi[k] * Binomial[n, k] * a[n - k], {k, 1, n}]; Array[a, 20, 0] (* _Amiram Eldar_, Apr 30 2022 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, eulerphi(k)*x^k/k))))

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (j-1)!*eulerphi(j)*binomial(i, j)*v[i-j+1])); v;

%Y Cf. A000010, A074930, A159929, A318917, A352887.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Apr 29 2022