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A133573 Expansion of ( 5 * phi(-q^5)^2 - phi(-q)^2 ) / 4 in powers of q where phi() is a Ramanujan theta function. 3
1, 1, -1, 0, -1, -3, 0, 0, -1, 1, 3, 0, 0, 2, 0, 0, -1, 2, -1, 0, 3, 0, 0, 0, 0, -7, -2, 0, 0, 2, 0, 0, -1, 0, -2, 0, -1, 2, 0, 0, 3, 2, 0, 0, 0, -3, 0, 0, 0, 1, 7, 0, -2, 2, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, -1, -6, 0, 0, -2, 0, 0, 0, -1, 2, -2, 0, 0, 0, 0, 0, 3, 1, -2, 0, 0, -6, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 2, -1, 0, 7, 2, 0, 0, -2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

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

Multiplicative because this sequence is the inverse Moebius transform of a multiplicative sequence. - Andrew Howroyd, Aug 06 2018

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

S. Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328; see p. 313 eq. (2.5).

L.-C. Shen, On the additive formulas of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5, Trans. Amer. Math. Soc. 345 (1994), no. 1, 323-345; see p. 335 eq. (3.5), p. 342 eq. (3.42).

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of eta(q^2)^3 * eta(q^5) / ( eta(q) * eta(q^10) ) in powers of q.

Euler transform of period 10 sequence [ 1, -2, 1, -2, 0, -2, 1, -2, 1, -2, ...].

Moebius transform is period 40 sequence [ 1, -2, -1, 0, -4, 2, -1, 0, 1, 8, -1, 0, 1, 2, 4, 0, 1, -2, -1, 0, 1, 2, -1, 0, -4, -2, -1, 0, 1, -8, -1, 0, 1, -2, 4, 0, 1, 2, -1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 20 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A122190.

a(n) = (-1)^n * A133574(n). a(2*n) = A133574(n). a(4*n + 1) = A214316(n). a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(9*n) = a(n). - Michael Somos, Jul 12 2012

EXAMPLE

1 + q - q^2 - q^4 - 3*q^5 - q^8 + q^9 + 3*q^10 + 2*q^13 - q^16 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (5 EllipticTheta[ 4, 0, q^5]^2 - EllipticTheta[ 4, 0, q]^2)/4, {q, 0, n}] (* Michael Somos, Jul 12 2012 *)

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^2 QPochhammer[ q^5, q^10] / QPochhammer[ q, q^2], {q, 0, n}] (* Michael Somos, Jul 12 2012 *)

PROG

(PARI) {a(n) = if( n<1, n==0, (-1)^n * sumdiv(n, d, if( d%5==0, kronecker(-4, d/5) * 5) - kronecker(-4, d)))}

(PARI) {a(n) = local(A); if( n<0, 0, A = x*O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^5+A) / (eta(x + A) * eta(x^10 + A)), n))}

CROSSREFS

Cf. A133574, A214316.

Sequence in context: A236322 A319419 A133574 * A151859 A163541 A280815

Adjacent sequences:  A133570 A133571 A133572 * A133574 A133575 A133576

KEYWORD

sign,mult

AUTHOR

Michael Somos, Sep 17 2007

STATUS

approved

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Last modified April 20 16:15 EDT 2021. Contains 343135 sequences. (Running on oeis4.)