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A328775 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} tau(n) * x^n, where tau = A000005.

%I A328775 #7 Oct 28 2019 11:51:06
%S A328775 1,2,0,2,-1,1,-1,2,1,-2,0,2,-1,-2,2,3,-2,-1,2,1,-4,0,3,-1,3,-3,-2,0,1,
%T A328775 9,-15,3,17,-13,-1,-1,9,-2,-18,27,-10,-14,24,-17,-15,24,27,-43,-37,72,
%U A328775 43,-116,-11,147,-98,-24,67,-56,24,-44,213,-258,-193,707,-435
%N A328775 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} tau(n) * x^n, where tau = A000005.
%C A328775 Inverse weigh transform of A000005.
%H A328775 Alois P. Heinz, <a href="/A328775/b328775.txt">Table of n, a(n) for n = 1..2500</a>
%p A328775 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A328775       add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A328775     end:
%p A328775 a:= proc(n) option remember; numtheory[tau](n)-b(n, n-1) end:
%p A328775 seq(a(n), n=1..80);  # _Alois P. Heinz_, Oct 27 2019
%t A328775 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = DivisorSigma[0, n] - b[n, n - 1]; Array[a, 65]
%Y A328775 Cf. A000005, A107742, A320779.
%K A328775 sign
%O A328775 1,2
%A A328775 _Ilya Gutkovskiy_, Oct 27 2019