A259743 Expansion of f(-x)^3 * psi(x^4) in powers of x where psi(), f() are Ramanujan theta functions.

1, -3, 0, 5, 1, -3, -7, 5, 0, 0, 2, 0, 1, -3, 9, -6, 0, 0, -7, -11, 0, 13, 9, 0, 1, 10, 0, -6, -15, 0, -7, 0, -15, 13, 9, 0, 17, 0, 0, -11, 3, -3, 0, 5, 0, -6, -7, 0, 17, -19, 9, 0, -15, 0, 0, 10, 0, -19, 0, 21, 18, 10, 0, 5, 0, 0, -30, 21, -15, -19, -14, 0

Offset

0,2

Comments

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

Links

Seiichi Manyama, 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^(-5/8) * eta(q)^3 * eta(q^8)^2 / eta(q^4) in powers of q.

Euler transform of period 8 sequence [ -3, -3, -3, -2, -3, -3, -3, -4, ...].

G.f.: Product_{k>0} (1 - x^k)^3 * (1 + x^(4*k)) * (1 - x^(8*k)).

Example

G.f. = 1 - 3*x + 5*x^3 + x^4 - 3*x^5 - 7*x^6 + 5*x^7 + 2*x^10 + x^12 + ...
G.f. = q^5 - 3*q^13 + 5*q^29 + q^37 - 3*q^45 - 7*q^53 + 5*q^61 + 2*q^85 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 EllipticTheta[ 2, 0, x^2] / (2 x^(1/2)) {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 QPochhammer[ x^8]^2 / QPochhammer[ x^4], {x, 0, n}];

Prog

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

Crossrefs

Sequence in context: A229979 A050925 A086696 * A247015 A241972 A357823

Adjacent sequences: A259740 A259741 A259742 * A259744 A259745 A259746

Keyword

sign

Author

Michael Somos, Jul 05 2015

Status

approved