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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. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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).

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

M. 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))}

CROSSREFS

Cf. A002291, A225912, A322032.

Sequence in context: A064821 A051239 A002291 * A231777 A288181 A110622

Adjacent sequences:  A225869 A225870 A225871 * A225873 A225874 A225875

KEYWORD

sign,changed

AUTHOR

Michael Somos, May 18 2013

STATUS

approved

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Last modified November 17 03:06 EST 2019. Contains 329216 sequences. (Running on oeis4.)