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A369494 a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} (x^(k*(k+1)/2) + 1/x^(k*(k+1)/2)).

%I A369494 #6 Jan 25 2024 08:03:34
%S A369494 1,1,0,0,0,2,0,3,3,5,0,14,23,39,0,101,161,315,0,971,1595,2872,0,9697,
%T A369494 17431,31736,0,103608,190242,356883,0,1218049,2235343,4165201,0,
%U A369494 14602056,27304610,51182196,0,179995388,339041695,640927871,0,2288387318,4326722468,8201714149
%N A369494 a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} (x^(k*(k+1)/2) + 1/x^(k*(k+1)/2)).
%p A369494 b:= proc(n, i) option remember; (m-> `if`(n>m, 0,
%p A369494      `if`(n=m, 1, b(abs(n-i*(i+1)/2), i-1)+
%p A369494         b(n+i*(i+1)/2, i-1))))((2+(3+i)*i)*i/6)
%p A369494     end:
%p A369494 a:= n-> `if`(irem(n, 4)=2, 0, b(n*(n+1)/2, n)):
%p A369494 seq(a(n), n=0..45);  # _Alois P. Heinz_, Jan 24 2024
%t A369494 Table[Coefficient[Product[x^(k (k + 1)/2) + 1/x^(k (k + 1)/2), {k, 1, n}], x, n (n + 1)/2], {n, 0, 45}]
%Y A369494 Cf. A000217, A063890, A158380, A351002, A368243.
%K A369494 nonn
%O A369494 0,6
%A A369494 _Ilya Gutkovskiy_, Jan 24 2024