OFFSET
0,2
COMMENTS
This is Glaisher's alpha(m) for odd values of m. - N. J. A. Sloane, Nov 24 2018
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 37).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/2) * k(q) * k'(q)^2 * (K(q) / (4 *(pi/2))^6) in powers of q where k(), k'(), K() are Jacobi elliptic functions.
Expansion of phi(-x^2)^8 * psi(x)^4 in powers of x where phi(), psi() are Ramanujan theta functions. (Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).)
Expansion of f(x)^8 * f(-x)^4 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-1/2) * (eta(q^2)^6 / (eta(q) * eta(q^4)^2))^4 in powers of q.
Euler transform of period 4 sequence [ 4, -20, 4, -12, ...].
|a(n)| = A002290(n).
EXAMPLE
1 + 4*x - 10*x^2 - 56*x^3 + 29*x^4 + 332*x^5 + 30*x^6 - 1064*x^7 + ...
or
q + 4*q^3 - 10*q^5 - 56*q^7 + 29*q^9 + 332*q^11 + 30*q^13 - 1064*q^15 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^4 QPochhammer[ -q]^8, {q, 0, n}]
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^4]^2 EllipticTheta[ 2, 0, q] / 2)^4, {q, 0, 1 + 2 n}]
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^6 / (eta(x + A) * eta(x^4 + A)^2))^4, n))}
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, May 17 2013
EXTENSIONS
Entry revised by N. J. A. Sloane, Apr 27 2014
STATUS
approved