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Revision History for A002448
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
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FORMULA
Form Peter Bala, Sep 27 2023: (Start)
G.f. A(xq) satisfies A(xq)*A(-xq) = A(xq^2)^2. - _Peter Bala_, Sep 27 2023
A(q) = Sum_{n >= 1} (-q)^(n-1)*Product_{k >= n} 1 - q^k. (End)
FORMULA
G.f: A(q) = eta(q^2)^5 / ( eta(-q)*eta(q^4) )^2.
A(-q)^2 = 1 + 4*Sum_{n >= 1} (-1)^(n+1)*q^(2*n-1)/(1 - q^(2*n-1)), which gives the number of representations of an integer as a sum of two squares. See, for example, Fine, 26.63.
G.f. A(x) satisfies A(x)*A(-x) = A(x^2)^2. - Peter Bala, Sep 27 2023
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PROG
(Python)
from sympy.ntheory.primetest import is_square
def A002448(n): return (-is_square(n) if n&1 else is_square(n))<<1 if n else 1 # Chai Wah Wu, May 17 2023
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