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A369344 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k*(k+1)/2) + 1 + 1/x^(k*(k+1)/2)).

%I A369344 #18 Jan 22 2024 04:45:51
%S A369344 1,1,1,1,3,5,11,27,61,133,311,761,1839,4575,11573,29641,76487,199617,
%T A369344 524067,1384697,3681069,9841217,26437741,71369101,193496241,526685793,
%U A369344 1438816755,3944034221,10845006963,29908325821,82707648985,229306378067,637283978821
%N A369344 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k*(k+1)/2) + 1 + 1/x^(k*(k+1)/2)).
%C A369344 All terms are odd.
%C A369344 a(n) is the number of solutions to 0 = Sum_{i=1..n} c_i * i*(i+1)/2 with c_i in {-1,0,1}.
%H A369344 Alois P. Heinz, <a href="/A369344/b369344.txt">Table of n, a(n) for n = 0..200</a>
%F A369344 a(n) ~ sqrt(5) * 3^(n + 1/2) / (sqrt(Pi) * n^(5/2)). - _Vaclav Kotesovec_, Jan 22 2024
%p A369344 b:= proc(n, i) option remember; `if`(n>i*(i+1)*(i+2)/6, 0, `if`(i=0, 1,
%p A369344       b(n, i-1)+b(n+i*(i+1)/2, i-1)+b(abs(n-i*(i+1)/2), i-1)))
%p A369344     end:
%p A369344 a:= n-> b(0, n):
%p A369344 seq(a(n), n=0..33);  # _Alois P. Heinz_, Jan 21 2024
%t A369344 Table[Coefficient[Product[x^(k (k + 1)/2) + 1 + 1/x^(k (k + 1)/2), {k, 1, n}], x, 0], {n, 0, 31}]
%Y A369344 Cf. A000217, A007576, A158380, A350249.
%K A369344 nonn
%O A369344 0,5
%A A369344 _Ilya Gutkovskiy_, Jan 20 2024