The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A002448 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A002448

Expansion of Jacobi theta function theta_4(x).
(history; published version)

#73 by Michael De Vlieger at Thu Sep 28 12:08:08 EDT 2023

STATUS

reviewed

approved

#72 by Joerg Arndt at Thu Sep 28 11:29:26 EDT 2023

STATUS

proposed

reviewed

#71 by Peter Bala at Thu Sep 28 10:35:52 EDT 2023

STATUS

editing

proposed

#70 by Peter Bala at Thu Sep 28 10:09:50 EDT 2023

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)

#69 by Peter Bala at Wed Sep 27 08:49:37 EDT 2023

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

STATUS

approved

editing

#68 by Joerg Arndt at Wed May 17 10:14:34 EDT 2023

STATUS

reviewed

approved

#67 by Michel Marcus at Wed May 17 09:27:34 EDT 2023

STATUS

proposed

reviewed

#66 by Chai Wah Wu at Wed May 17 09:26:18 EDT 2023

STATUS

editing

proposed

#65 by Chai Wah Wu at Wed May 17 09:26:11 EDT 2023

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

STATUS

approved

editing

#64 by Susanna Cuyler at Wed Feb 24 08:22:34 EST 2021

STATUS

proposed

approved