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A305881 Expansion of Product_{k>=1} 1/(1 + prime(k)*x^k).

%I A305881 #9 Jun 15 2018 10:28:07
%S A305881 1,-2,1,-7,16,-28,62,-118,303,-630,1152,-2426,5315,-10718,20482,
%T A305881 -43449,91111,-179254,358910,-727829,1484601,-2995681,5924606,
%U A305881 -11935441,24382120,-48702245,96682698,-195063604,392983826,-784903199,1569490057,-3146479152,6317124649,-12652202092
%N A305881 Expansion of Product_{k>=1} 1/(1 + prime(k)*x^k).
%C A305881 Convolution inverse of A147655.
%H A305881 Alois P. Heinz, <a href="/A305881/b305881.txt">Table of n, a(n) for n = 0..3321</a>
%F A305881 G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*prime(j)^k*x^(j*k)/k).
%p A305881 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A305881       b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*ithprime(i))))
%p A305881     end:
%p A305881 a:= proc(n) option remember; `if`(n=0, 1,
%p A305881       -add(b(n-i$2)*a(i$2), i=0..n-1))
%p A305881     end:
%p A305881 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jun 13 2018
%t A305881 nmax = 33; CoefficientList[Series[Product[1/(1 + Prime[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A305881 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 A305881 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 A305881 Cf. A000040, A002099, A061151, A145519, A147655, A298160, A304791, A305882.
%K A305881 sign
%O A305881 0,2
%A A305881 _Ilya Gutkovskiy_, Jun 13 2018