A225872 Expansion of k(q)^3 * k'(q)^2 * (K(q) / (Pi/2))^6 / 64 in powers of q where k(), k'(), K() are Jacobi elliptic functions.

0, 1, -4, 2, 8, -13, 28, -26, -56, 69, -48, 134, 80, -182, -84, -312, 280, 204, 332, 142, -816, 91, -196, 780, -224, -526, -244, -1198, 2216, 767, 508, -390, -400, -1167, -1424, 466, -2264, 1391, 1392, 3796, -1480, -11, 1768, -2274, 1320, -1508, -1984, -8450

Offset: 0

Author

Michael Somos, May 18 2013

Keywords

• sign

Comments

In Glaisher (1907) this is denoted by beta'(m) = beta(m)/16 on page 56 while beta(m) (see A322032) is defined on page 38.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			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 + ...
		

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

Crossrefs

Cf. A002291, A225912, A322032.

Programs

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}]

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))}

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).