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A350495 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1/x^(k^2))^2.

%I A350495 #61 Feb 05 2022 05:41:21
%S A350495 1,2,4,8,16,40,88,222,570,1564,4516,13874,41866,137432,442964,1492610,
%T A350495 4998674,17204844,59175316,207299554,727137516,2582078416,9179001124,
%U A350495 32943918428,118453240846,428937325964,1556421977612,5676923326262,20754245720206
%N A350495 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1/x^(k^2))^2.
%H A350495 Vaclav Kotesovec, <a href="/A350495/b350495.txt">Table of n, a(n) for n = 0..200</a>
%F A350495 Conjecture: a(n) ~ sqrt(5) * 4^n / (sqrt(Pi) * n^(5/2)). - _Vaclav Kotesovec_, Feb 05 2022
%p A350495 b:= proc(n) option remember; `if`(n=0, 1,
%p A350495       expand((x^(n^2)+1/x^(n^2))^2*b(n-1)))
%p A350495     end:
%p A350495 a:= n-> coeff(b(n),x,0):
%p A350495 seq(a(n), n=0..28);  # _Alois P. Heinz_, Jan 28 2022
%t A350495 Table[Coefficient[Product[(x^(k^2) + 1/x^(k^2))^2, {k, 1, n}], x, 0], {n, 0, 30}] (* _Vaclav Kotesovec_, Feb 05 2022 *)
%Y A350495 Cf. A000980, A047653, A158092, A350249, A350881.
%K A350495 nonn
%O A350495 0,2
%A A350495 _Ilya Gutkovskiy_, Jan 28 2022