Reciprocal Of Some Euler Sums

[Leroy Quet]

I posted the message below to sci.math, with no responses.

Maybe someone can extend the sequence.

Thanks.

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I was wondering about the Dirichlet sum:

sum{k=1 to oo} a(k)/k^r =

1/(sum{k=1 to oo} (sum{j=1 to k} 1/j) /k^r)

= 
(for r =integer >= 2)

1/(zeta(r+1)(r/2 +1) -(1/2)sum{j=2 to r-1} zeta(j)zeta(r+1-j))

We have a(1) = 1,

and, for m >= 2,

a(m) = -sum{k|m,k>=2} (sum{j=1 to k} 1/j) a(m/k),

where sum{j=1 to k} 1/j  = H(k), the k-th harmonic number.

We have the a-sequence (if I did not err):
1, -3/2, -11/6, 1/6, -137/60, 61/20,...

What is the closed-form (non-recursive form) for the terms of this 
sequence?

thanks,
Leroy Quet