A134178 - OEIS

A134178

Expansion of chi(x) * chi(-x^2)^2 * chi(-x^3) * chi(-x^4) * chi(x^6)^2 * chi(-x^12) in powers of x where chi() is a Ramanujan theta function.

1, 1, -2, -2, 0, 1, 2, 0, 0, -1, -4, 0, 1, 0, 6, 2, 0, 1, -8, 0, 0, 0, 12, 0, -1, -1, -18, -4, 0, -1, 24, 0, 0, 2, -32, 0, 0, 1, 44, 6, 0, -2, -58, 0, 0, -1, 76, 0, 1, 2, -100, -8, 0, 1, 128, 0, 0, -3, -164, 0, 0, -1, 210, 12, 0, 4, -264, 0, 0, 2, 332, 0, -1, -5, -416, -18, 0, -2, 516, 0, 0, 5, -640, 0, -1, 2, 790, 24

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OFFSET

0,3

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Euler transform of period 24 sequence [ 1, -3, 0, -1, 1, -1, 1, 0, 0, -3, 1, -4, 1, -3, 0, 0, 1, -1, 1, -1, 0, -3, 1, 0, ...].

a(12*n + 4) = a(12*n + 7) = a(12*n + 8) = a(12*n + 11) = 0.

a(4*n + 1) = a(12*n) = A029838(n). a(4*n + 2) = a(12*n + 3) = -2 * A083365(n).

EXAMPLE

G.f. = 1 + x - 2*x^2 - 2*x^3 + x^5 + 2*x^6 - x^9 - 4*x^10 + x^12 + 6*x^14 + ...

G.f. = q^-3 + q^-1 - 2*q - 2*q^3 + q^7 + 2*q^9 - q^15 - 4*q^17 + q^21 + 6*q^25 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^2, x^4]^2 QPochhammer[ x^3, x^6] QPochhammer[ x^4, x^8] QPochhammer[-x^6, x^12]^2 QPochhammer[ x^12, x^24], {x, 0, n}]; (* Michael Somos, Oct 25 2015 *)

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^12, x^24] QPochhammer[ x^24, x^48] + x QPochhammer[ -x^4, x^8] QPochhammer[ x^8, x^16] - 2 x^2 QPochhammer[ x^16] / QPochhammer[ -x^4] - 2 x^3 QPochhammer[ x^48] / QPochhammer[ -x^12], {x, 0, n}]; (* Michael Somos, Oct 25 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A) * eta(x^12 + A)^5 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^3 * eta(x^8 + A) * eta(x^24 + A)^3), n))};

CROSSREFS

Cf. A029838, A083365.

Sequence in context: A104579 A079531 A182882 * A059018 A249371 A122190

Adjacent sequences: A134175 A134176 A134177 * A134179 A134180 A134181

KEYWORD

sign

AUTHOR

Michael Somos, Oct 11 2007

STATUS

approved