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

Author

Ilya Gutkovskiy, Oct 03 2019

Keywords

• sign

Comments

Logarithmic transform of A000010.

Links

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

Crossrefs

Cf. A000010, A300011, A318811, A318917.

Programs

Maple

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

Mathematica

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}]

Formula

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