OFFSET
0,3
COMMENTS
REFERENCES
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of x * (psi(x) * psi(-x)^2)^4 in powers of x where psi() is a Ramanujan theta function.
Expansion of x * (f(-x) * f(-x^4)^2)^4 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-1/2) * (eta(q) * eta(q^4)^2)^4 in powers of q.
Euler transform of period 4 sequence [ -4, -4, -4, -12, ...].
G.f. is a period 1 Fourier series which satisfies f(-1/(8*t)) = 16 * (t/i)^6 * g(t) where q = exp(2*Pi*i*t) and g() is the g.f. for A225912.
G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(4*k))^2)^4.
|a(n)| = A002291(n).
EXAMPLE
x - 4*x^2 + 2*x^3 + 8*x^4 - 13*x^5 + 28*x^6 - 26*x^7 - 56*x^8 + 69*x^9 + ...
q^3 - 4*q^5 + 2*q^7 + 8*q^9 - 13*q^11 + 28*q^13 - 26*q^15 - 56*q^17 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^4]^2)^4, {q, 0, n}]
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, I q]^2 )^4 / -4096, {q, 0, 2 n + 1}]
PROG
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2)^4, n))}
KEYWORD
sign
AUTHOR
Michael Somos, May 18 2013
STATUS
approved