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A307522 Expansion of Product_{k>=1} ((1 + x)^k - x^k)/((1 + x)^k + x^k).

%I A307522 #23 Apr 12 2025 06:29:50
%S A307522 1,-2,2,-2,4,-10,22,-42,72,-116,188,-332,662,-1432,3148,-6736,13784,
%T A307522 -26894,50254,-90782,160856,-285230,518170,-983710,1964800,-4090002,
%U A307522 8705322,-18582722,39219572,-81148034,163946630,-323136562,622125982,-1173528562,2179230066
%N A307522 Expansion of Product_{k>=1} ((1 + x)^k - x^k)/((1 + x)^k + x^k).
%F A307522 G.f.: theta_4(x/(1 + x)), where theta_4() is the Jacobi theta function.
%F A307522 From _Peter Bala_, Dec 31 2024: (Start)
%F A307522 For n >= 1, a(n) = 2 * (-1)^n * Sum_{k = 1..floor(sqrt(n))} binomial(n-1, n-k^2).
%F A307522 For n >= 1, |a(n)| = 2 * A103198(n). (End)
%p A307522 a(n) := 2*(-1)^n*add( binomial(n-1, n-k^2), k = 1..floor(sqrt(n))):
%p A307522 print(1, seq(a(n), n = 1..40)); # _Peter Bala_, Dec 31 2024
%t A307522 m = 34; CoefficientList[Series[Product[((1 + x)^k - x^k)/((1 + x)^k + x^k), {k, 1, m}], {x, 0, m}], x] (* _Amiram Eldar_, May 14 2021 *)
%o A307522 (PARI) my(N=66, x='x+O('x^N)); Vec(prod(k=1, N, ((1+x)^k-x^k)/((1+x)^k+x^k)))
%Y A307522 Cf. A002448, A103198, A307520, A307521, A318570.
%K A307522 sign,easy
%O A307522 0,2
%A A307522 _Seiichi Manyama_, Apr 12 2019