A262158 -id:A262158

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A262929

Expansion of phi(-x^3) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

+10
3

1, 0, 0, -2, -1, 0, 0, 2, 1, 0, 0, -2, 0, 0, 0, 4, 1, 0, 0, -6, -2, 0, 0, 8, 1, 0, 0, -12, -1, 0, 0, 16, 2, 0, 0, -22, -3, 0, 0, 30, 2, 0, 0, -38, -1, 0, 0, 50, 4, 0, 0, -66, -5, 0, 0, 84, 3, 0, 0, -106, -3, 0, 0, 136, 6, 0, 0, -172, -8, 0, 0, 214, 5, 0, 0

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OFFSET

0,4

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(1/2) * eta(q^3)^2 * eta(q^4) / (eta(q^6) * eta(q^8)^2) in powers of q.

Euler transform of period 24 sequence [0, 0, -2, -1, 0, -1, 0, 1, -2, 0, 0, -2, 0, 0, -2, 1, 0, -1, 0, -1, -2, 0, 0, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = (32/3)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261877.

a(4*n) = A143066(n). a(4*n + 1) = a(4*n + 2) = 0. a(4*n + 3) = -2 * A262160(n).

a(12*n) = A262150(n). a(12*n + 3) = -2*A262151(n). a(12*n + 4) = -A262152(n). a(12*n + 7) = 2*A262156(n). a(12*n + 8) = A262157(n). a(12*n + 11) = -2*A262158(n). - Michael Somos, Apr 03 2016

Convolution inverse is A261877. - Michael Somos, Oct 22 2017

EXAMPLE

G.f. = 1 - 2*x^3 - x^4 + 2*x^7 + x^8 - 2*x^11 + 4*x^15 + x^16 + ...

G.f. = q^-1 - 2*q^5 - q^7 + 2*q^13 + q^15 - 2*q^21 + 4*q^29 + q^31 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ 2 x^(1/2) EllipticTheta[ 4, 0, x^3] / EllipticTheta[ 2, 0, x^2], {x, 0, n}];

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A) / (eta(x^6 + A) * eta(x^8 + A)^2), n))};

(PARI) q='q+O('q^99); Vec(eta(q^3)^2*eta(q^4)/(eta(q^6)*eta(q^8)^2)) \\ Altug Alkan, Jul 31 2018

CROSSREFS

Cf. A143066, A261877, A262150, A262151, A262152, A262156, A262157, A262158, A262160.

KEYWORD

sign

AUTHOR

Michael Somos, Oct 04 2015

STATUS

approved

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