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A339435 G.f.: Sum_{k>=0} (-1)^k * k! * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

%I A339435 #8 Dec 05 2020 21:14:52
%S A339435 1,-1,-1,1,1,3,-3,-1,-7,-11,7,3,15,35,71,-35,25,-57,-99,-277,-415,25,
%T A339435 -185,39,327,1079,1895,3745,-71,2907,813,479,-4927,-7259,-20393,
%U A339435 -29877,-8409,-28621,-23041,-16811,4441,27783,102741,169595,324065,105265,361471,280983,385215
%N A339435 G.f.: Sum_{k>=0} (-1)^k * k! * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).
%C A339435 The difference between the number of compositions (ordered partitions) of n into an even number of distinct parts and the number of compositions (ordered partitions) of n into an odd number of distinct parts.
%H A339435 Alois P. Heinz, <a href="/A339435/b339435.txt">Table of n, a(n) for n = 0..5000</a>
%F A339435 a(n) = A332305(n) - A332304(n).
%p A339435 b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, p!
%p A339435       *(1-2*irem(p, 2)), add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
%p A339435     end:
%p A339435 a:= n-> b(n$2, 0):
%p A339435 seq(a(n), n=0..55);  # _Alois P. Heinz_, Dec 04 2020
%t A339435 nmax = 48; CoefficientList[Series[Sum[(-1)^k k! x^(k (k + 1)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%Y A339435 Cf. A010815, A032020, A081362, A332304, A332305.
%K A339435 sign,look
%O A339435 0,6
%A A339435 _Ilya Gutkovskiy_, Dec 04 2020