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A300508 Expansion of Product_{k>=1} (1 - x^k)^p(k), where p(k) = number of partitions of k (A000041).

%I A300508 #8 Mar 11 2018 20:07:33
%S A300508 1,-1,-2,-1,-1,3,3,9,9,10,8,-1,-21,-45,-77,-130,-163,-198,-179,-108,
%T A300508 101,451,1058,1878,2999,4276,5595,6511,6446,4443,-838,-11069,-28373,
%U A300508 -54652,-91948,-140370,-198501,-259706,-311997,-332003,-285486,-118600,239086,881998,1918851,3470261
%N A300508 Expansion of Product_{k>=1} (1 - x^k)^p(k), where p(k) = number of partitions of k (A000041).
%C A300508 Convolution inverse of A001970.
%H A300508 Alois P. Heinz, <a href="/A300508/b300508.txt">Table of n, a(n) for n = 0..3000</a>
%F A300508 G.f.: Product_{k>=1} (1 - x^k)^A000041(k).
%p A300508 with(numtheory): with(combinat):
%p A300508 b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A300508       numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
%p A300508     end:
%p A300508 a:= proc(n) option remember; `if`(n=0, 1,
%p A300508       -add(b(n-i)*a(i), i=0..n-1))
%p A300508     end:
%p A300508 seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 07 2018
%t A300508 nmax = 45; CoefficientList[Series[Product[(1 - x^k)^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]
%Y A300508 Cf. A000041, A001970, A261049.
%K A300508 sign
%O A300508 0,3
%A A300508 _Ilya Gutkovskiy_, Mar 07 2018