A145702 Expansion of chi(-x) * chi(x^5) in powers of x where chi() is a Ramanujan theta function.

1, -1, 0, -1, 1, 0, 0, -1, 1, -1, 1, -1, 2, -1, 1, -1, 2, -2, 1, -2, 3, -3, 2, -3, 4, -3, 2, -4, 5, -4, 4, -5, 6, -6, 5, -6, 8, -7, 6, -8, 11, -10, 8, -11, 13, -11, 10, -13, 16, -15, 14, -17, 20, -18, 17, -20, 24, -23, 21, -25, 31, -29, 26, -32, 37, -34, 32

Offset

0,13

Comments

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

Links

Table of n, a(n) for n=0..66.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(1/4) * eta(q) * eta(q^10)^2 / eta(q^2) / eta(q^5) / eta(q^20) in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A145703.

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

a(n) = (-1)^n * A139632(n). a(2*n) = A139631(n). a(2*n + 1) = - A145703(n).

a(n) = -(-1)^floor(n/2) * A145704(n) = (-1)^floor((n + 1)/2) * A145705(n). - Michael Somos, Sep 06 2015

Example

G.f. = 1 - x - x^3 + x^4 - x^7 + x^8 - x^9 + x^10 - x^11 + 2*x^12 - x^13 + ...
G.f. = 1/q - q^3 - q^11 + q^15 - q^27 + q^31 - q^35 + q^39 - q^43 + 2*q^47 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^5, x^10], {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 + A) * eta(x^10 + A)^2 / eta(x^2 + A) / eta(x^5 + A) / eta(x^20 + A), n))};

Crossrefs

Cf. A139631, A139632, A147503, A145704, A145705.

Sequence in context: A229873 A135230 A117957 * A145704 A139632 A145705

Adjacent sequences: A145699 A145700 A145701 * A145703 A145704 A145705

Keyword

sign

Author

Michael Somos, Oct 17 2008

Status

approved