A322513 Expansion of e.g.f. log(1 + Sum_{k>=1} d(k) * x^k / k!), where d(k) = number of divisors of k (A000005).

0, 1, 1, -2, 1, 11, -48, -6, 1241, -6431, -15320, 452970, -2317212, -17584137, 372119776, -1552313624, -31732274313, 565880016193, -1217992446564, -90197542736656, 1400682677566587, 1990004001731140, -384348195167184028, 5109122826021406702

Offset: 0

Author

Ilya Gutkovskiy, Oct 03 2019

Keywords

• sign

Comments

Logarithmic transform of A000005.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..470

Crossrefs

Cf. A000005, A028342, A274805, A294363, A295739.

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[tau]))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 06 2019

Mathematica

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