A107033 Expansion of f(x, x) * f(x, -x^2) in powers of x where f(,) is a Ramanujan theta function.

1, 3, 1, -2, 2, 1, -4, -1, -2, 0, 2, -4, -1, -2, -2, 1, 0, 2, -2, 2, 0, -4, 1, 0, 2, 2, 5, 0, -2, 0, 0, 4, -2, 0, 0, 3, 4, 0, 0, 2, 1, -4, 2, -2, 0, 0, 0, 2, -2, 0, 2, 3, -2, 0, -2, -2, -4, -1, 0, 0, 0, -4, 2, 0, 4, 0, -4, -2, 0, -2, -1, 0, 0, -2, -2, 2, -6, 1, 2, 0, 0, 4, 0, -2, 2, 0, 0, -2, -2, -2, 2, 0, 1, 0, 0, -2, 4, 0, 0, 2, 1, 6, 0, 2, 0

Offset

0,2

Comments

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

References

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339, see page 333.

H. Kahl, G. Koehler, Components of Hecke theta series, J. Math. Anal. Appl. 232 (1999), no. 2, 312-331, see page 320. MR1683136 (2000e:11051)

Links

Table of n, a(n) for n=0..104.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(sqrt(-x)) * f(-sqrt(-x)) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Aug 23 2010

Expansion of f(x) * phi(x) in powers of x where f() and phi() are Ramanujan theta functions. - Michael Somos, Aug 23 2010

Expansion of q^(-1/24) * eta(q^2)^8 / (eta(q)^3 * eta(q^4)^3) in powers of q.

Euler transform of period 4 sequence [ 3, -5, 3, -2, ...].

G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^k)^3 / (1 + x^(2*k))^3.

a(n) = (-1)^n * A115110(n).

Example

1 + 3*x + x^2 - 2*x^3 + 2*x^4 + x^5 - 4*x^6 - x^7 - 2*x^8 + 2*x^10 + ...
q + 3*q^25 + q^49 - 2*q^73 + 2*q^97 + q^121 - 4*q^145 - q^169 - 2*q^193 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ I x] QPochhammer[ -I x], {x, 0, 2 n}] (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x] EllipticTheta[ 3, 0, x], {x, 0, n}] (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^2 QPochhammer[ -x, x]^3 / QPochhammer[ -x^2, x^2]^3, {x, 0, n}] (* Michael Somos, Jul 12 2012 *)

Prog

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

Crossrefs

Cf. A115110.

Sequence in context: A296518 A111951 A222593 * A115110 A066635 A016568

Adjacent sequences: A107030 A107031 A107032 * A107034 A107035 A107036

Keyword

sign

Author

Michael Somos, May 09 2005

Status

approved