A145706 - OEIS

A145706

Expansion of chi(-x^5) / chi(-x^2) in powers of x where chi() is a Ramanujan theta function.

1, 0, 1, 0, 1, -1, 2, -1, 2, -1, 3, -2, 4, -2, 5, -4, 6, -5, 8, -6, 11, -8, 13, -10, 16, -14, 20, -17, 24, -21, 31, -26, 37, -32, 44, -41, 54, -49, 64, -59, 79, -72, 94, -86, 111, -106, 132, -126, 156, -149, 187, -178, 219, -210, 257, -251, 302, -295, 352

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OFFSET

0,7

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

Expansion of q^(1/8) * eta(q^4) * eta(q^5) / (eta(q^2) * eta(q^10)) in powers of q.

Euler transform of period 20 sequence [ 0, 1, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 1, 0, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (1280 t)) = f(t) where q = exp(2 Pi i t).

G.f.: Product_{k>0} (1 - x^(10*k - 5)) / (1 - x^(4*k - 2)).

a(n) = (-1)^n * A139631(n) = A145704(2*n) = A145705(2*n).

EXAMPLE

G.f. = 1 + x^2 + x^4 - x^5 + 2*x^6 - x^7 + 2*x^8 - x^9 + 3*x^10 - 2*x^11 + ...

G.f. = 1/q + q^15 + q^31 - q^39 + 2*q^47 - q^55 + 2*q^63 - q^71 + 3*q^79 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ x^5, x^10] QPochhammer[ -x^2, x^2], {x, 0, n}]; (* Michael Somos, Sep 06 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^5 + A) / (eta(x^2 + A) * eta(x^10 + A)), n))};

CROSSREFS

Cf. A139631, A145704, A145705.

Sequence in context: A154958 A025806 A025802 * A139631 A029177 A321298

Adjacent sequences: A145703 A145704 A145705 * A145707 A145708 A145709

KEYWORD

sign

AUTHOR

Michael Somos, Oct 17 2008, Oct 20 2008

STATUS

approved