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%I #11 May 08 2022 08:44:56
%S 3,5,-8,35,-52,118,-320,1597,-2016,6616,-16064,40516,-122552,381606,
%T -903176,4389807,-7597004,22835416,-61172890,188526110,-486889660,
%U 1550995910,-4093173788,11608277912,-33815484714,105179650108,-279683446078,883705997682,-2366564864546
%N Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} prime(n+1)*x^n.
%t nn = 29; f[x_] := Product[(1 + a[n] x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Sum[Prime[k + 1] x^k, {k, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
%Y Cf. A006005, A065091, A147557, A353606.
%K sign
%O 1,1
%A _Ilya Gutkovskiy_, May 07 2022