A318811 -id:A318811 - cp's OEIS frontend

A328053 Expansion of e.g.f. log(1 + Sum_{k>=1} phi(k) * x^k / k!), where phi = Euler totient function (A000010).

0, 1, 0, 1, -3, 8, -32, 166, -926, 5842, -42812, 348632, -3088388, 29871372, -314102574, 3554714938, -43057252520, 556487433400, -7644034688586, 111160926400032, -1706191876272876, 27567942738717360, -467712309003533398, 8312805777830133096

Offset: 0

Ilya Gutkovskiy, Oct 03 2019

sign

Logarithmic transform of A000010.

Vaclav Kotesovec, Table of n, a(n) for n = 0..465

Cf. A000010, A300011, A318811, A318917.

a:= proc(n) option remember; `if`(n=0, 0, (b-> b(n)-add(a(j)
     *binomial(n, j)*j*b(n-j), j=1..n-1)/n)(numtheory[phi]))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 06 2019

nmax = 23; CoefficientList[Series[Log[1 + Sum[EulerPhi[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = EulerPhi[n] - Sum[Binomial[n, k] EulerPhi[n - k] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 23}]

a(n)/n! ~ -(-1)^n * d^n / n, where d = 0.8078801380543809482705136550646927880437760099284517780830096910529492372472... - Vaclav Kotesovec, Oct 17 2019

A352887 Expansion of e.g.f. 1/(1 - Sum_{k>=1} phi(k)*x^k/k!), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 3, 14, 84, 634, 5740, 60626, 731852, 9938670, 149966116, 2489148386, 45070961740, 884107377360, 18676602726734, 422721143355808, 10205605681874952, 261789688633794528, 7110331886095458918, 203848868169846041430, 6151813078359073154568

Offset: 0

Seiichi Manyama, Apr 07 2022

nonn

Cf. A000010, A159929, A300011, A318811.

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, eulerphi(k)*x^k/k!))))

a(n) = if(n==0, 1, sum(k=1, n, eulerphi(k)*binomial(n, k)*a(n-k)));

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

cp's OEIS Frontend

A328053 Expansion of e.g.f. log(1 + Sum_{k>=1} phi(k) * x^k / k!), where phi = Euler totient function (A000010).

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