A083650 Expansion of f(-x, x^3) * phi(x^2) in powers of x where phi(), f() are Ramanujan theta functions.

1, -1, 2, -1, 0, 2, -1, 0, 0, -2, -1, 2, -2, 0, -2, 1, 0, 2, 0, -2, 0, 1, 0, 0, -2, 0, 0, 0, -1, -2, 2, 0, 2, 0, 0, 2, 3, 0, 0, -2, 0, 0, -2, 0, 2, -1, 2, 0, 0, 0, 2, -2, 0, -2, 2, 1, -2, 2, 0, 0, 0, 0, 0, 0, 0, 2, -1, 0, 0, 0, 0, -2, 2, 0, -2, 2, 0, -2, -1, 0, -2, 0, -2, 0, -2, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, -2, -2, 0, 0, 0, 2, 2, 0, 0, -2

Offset

0,3

Comments

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

Essentially the expansion of eta(q)*eta(q^2). Cf. A010815. - N. J. A. Sloane, Feb 18 2010

A030204, A083650 and A138514 are the same except for signs. - N. J. A. Sloane, May 07 2010

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 16 sequence [ -1, 2, 1, -2, 1, 1, -1, -3, -1, 1, 1, -2, 1, 2, -1, -2, ...].

G.f.: (Sum_{k>=0} (-1)^(k + [k/4]) * x^(k*(k+1)/2)) * (Sum_k x^(2*k^2)).

(-1)^[n/2] * a(n) = A030204(n).

Example

G.f. = 1 - x + 2*x^2 - x^3 + 2*x^5 - x^6 - 2*x^9 - x^10 + 2*x^11 - 2*x^12 - 2*x^14 + ...
G.f. = q - q^9 + 2*q^17 - q^25 + 2*q^41 - q^49 + 2*q^73 - q^81 + 2*q^89 - 2*q^97 + ...

Mathematica

QP = QPochhammer; s = QP[q]*QP[q^2] + O[q]^105; Table[(-1)^Quotient[n, 2]*Coefficient[s, q, n], {n, 0, 105}] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); (-1)^(n\2) * polcoeff( eta(x + A) * eta(x^2 + A), n))}; /* Michael Somos, Mar 02 2010 */

Crossrefs

Cf. A000122, A000700, A010054, A010815, A030204, A083650, A121373, A138514, A143433.

Sequence in context: A156319 A190893 A030204 * A138514 A286137 A320658

Adjacent sequences: A083647 A083648 A083649 * A083651 A083652 A083653

Keyword

sign

Author

Michael Somos, May 01 2003

Extensions

Revised by Michael Somos, Mar 02 2010

Status

approved