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Expansion of Product_{n >= 1} (1 - x^(2*n))*(1 - x^(2*n-1))*(1 - x^(2*n+1)).
2

%I #15 Jan 03 2021 01:47:27

%S 1,-1,-1,-1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,-1,-1,-1,

%T -1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,

%U -1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1

%N Expansion of Product_{n >= 1} (1 - x^(2*n))*(1 - x^(2*n-1))*(1 - x^(2*n+1)).

%C The sequence consists of a 1, followed by three negative ones, followed by five ones, followed by seven negative ones, and so on.

%F O.g.f.: A(x) = theta_4(x)/(1 - x) = 1/(1 - x) * Sum_{n >= 0 } (-1)^n*x^(n^2), where theta_4(x) is the Jacobi theta function - see A002448. Note 1/A(x) is the generating function for A211971.

%p series( (1 + 2*add((-1)^n*x^(n^2), n = 1..10))/(1 - x), x, 101);

%Y Cf. A002448, A211971, A255175 (partial sums), A329116 (partial sums).

%K sign,easy

%O 0

%A _Peter Bala_, Nov 29 2020