%I #9 Jun 15 2018 10:28:07

%S 1,-2,1,-7,16,-28,62,-118,303,-630,1152,-2426,5315,-10718,20482,

%T -43449,91111,-179254,358910,-727829,1484601,-2995681,5924606,

%U -11935441,24382120,-48702245,96682698,-195063604,392983826,-784903199,1569490057,-3146479152,6317124649,-12652202092

%N Expansion of Product_{k>=1} 1/(1 + prime(k)*x^k).

%C Convolution inverse of A147655.

%H Alois P. Heinz, <a href="/A305881/b305881.txt">Table of n, a(n) for n = 0..3321</a>

%F G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*prime(j)^k*x^(j*k)/k).

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*ithprime(i))))

%p end:

%p a:= proc(n) option remember; `if`(n=0, 1,

%p -add(b(n-i$2)*a(i$2), i=0..n-1))

%p end:

%p seq(a(n), n=0..40); # _Alois P. Heinz_, Jun 13 2018

%t nmax = 33; CoefficientList[Series[Product[1/(1 + Prime[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 33; CoefficientList[Series[Exp[Sum[Sum[(-1)^k Prime[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (-Prime[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

%Y Cf. A000040, A002099, A061151, A145519, A147655, A298160, A304791, A305882.

%K sign

%O 0,2

%A _Ilya Gutkovskiy_, Jun 13 2018