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A329157 Expansion of Product_{k>=1} (1 - Sum_{j>=1} j * x^(k*j)).

%I A329157 #22 Jul 30 2025 17:26:09
%S A329157 1,-1,-3,-3,-4,3,2,19,21,32,40,45,16,8,-18,-125,-164,-291,-358,-530,
%T A329157 -588,-724,-592,-675,-358,-207,570,1201,2208,3333,4944,6490,8277,
%U A329157 10492,11800,13260,14328,14722,12942,12075,5640,603,-10444,-21120,-39360,-55876,-83488
%N A329157 Expansion of Product_{k>=1} (1 - Sum_{j>=1} j * x^(k*j)).
%C A329157 Convolution inverse of A329156.
%H A329157 Alois P. Heinz, <a href="/A329157/b329157.txt">Table of n, a(n) for n = 0..10000</a>
%F A329157 G.f.: Product_{k>=1} (1 - x^k / (1 - x^k)^2).
%F A329157 G.f.: exp(-Sum_{k>=1} ( Sum_{d|k} 1 / (d * (1 - x^(k/d))^(2*d)) ) * x^k).
%F A329157 G.f.: Product_{k>=1} (1 - x^k)^A032198(k).
%F A329157 G.f.: A(x) = Product_{k>=1} 1 / B(x^k), where B(x) = g.f. of A088305.
%F A329157 a(n) = Sum_{k=0..A003056(n)} (-1)^k * A385001(n,k). - _Alois P. Heinz_, Jul 18 2025
%p A329157 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1, b(n, i-1), 0)-
%p A329157       add(b(n-i*j, min(n-i*j, i-1))*j, j=`if`(i=1, n, 1..n/i)))
%p A329157     end:
%p A329157 a:= n-> b(n$2):
%p A329157 seq(a(n), n=0..46);  # _Alois P. Heinz_, Jul 18 2025
%t A329157 nmax = 46; CoefficientList[Series[Product[(1 - Sum[j x^(k j), {j, 1, nmax}]), {k, 1, nmax}], {x, 0, nmax}], x]
%t A329157 nmax = 46; CoefficientList[Series[Product[(1 - x^k/(1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A329157 Cf. A003056, A032198, A077285, A088305, A104575, A319668, A329156, A385001.
%K A329157 sign
%O A329157 0,3
%A A329157 _Ilya Gutkovskiy_, Nov 06 2019