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A128583 Expansion of chi(x) * psi(x^2) * phi(-x^6) in powers of x where phi(), psi(), chi() are Ramanujan theta functions. 7
1, 1, 1, 2, 1, 2, 1, 1, 1, 0, 3, 1, 1, 1, 2, 2, 1, 2, 0, 1, 2, 1, 0, 1, 2, 3, 0, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 0, 2, 1, 2, 0, 1, 0, 1, 4, 1, 2, 0, 1, 2, 1, 2, 1, 1, 3, 0, 1, 2, 3, 1, 0, 1, 0, 0, 2, 2, 1, 1, 2, 2, 1, 1, 2, 0, 1, 2, 0, 1, 1, 6, 1, 1, 1, 0, 2, 1 (list; graph; refs; listen; history; text; internal format)
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..1000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(-5/24) * eta(q^2) * eta(q^4) * eta(q^6)^2 / (eta(q) * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [ 1, 0, 1, -1, 1, -2, 1, -1, 1, 0, 1, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 6^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229723.

a(n) = A128582(4*n) = A259895(3*n) = A260118(4*n). 2 * a(n) = A190615(12*n + 2). - Michael Somos, Nov 15 2015

-2 * a(n) = A128580(12*n + 2). - Michael Somos, Dec 22 2016

EXAMPLE

G.f. = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + x^6 + x^7 + x^8 + 3*x^10 + ...

G.f. = q^5 + q^29 + q^53 + 2*q^77 + q^101 + 2*q^125 + q^149 + q^173 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^4] QPochhammer[ x^6]^2 / (QPochhammer[ x] QPochhammer[ x^12] ), {x, 0, n}];

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 4, 0, x^6] EllipticTheta[ 2, 0, x] / (2 x^(1/4)), {x, 0, n}];

PROG

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

CROSSREFS

Cf. A128580, A128582, A190615, A229723, A259895, A260118.

Sequence in context: A232624 A287368 A108149 * A218854 A172303 A064391

Adjacent sequences:  A128580 A128581 A128582 * A128584 A128585 A128586

KEYWORD

nonn

AUTHOR

Michael Somos, Mar 11 2007

STATUS

approved

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Last modified August 7 19:57 EDT 2020. Contains 336279 sequences. (Running on oeis4.)