A213622 Expansion of phi(x) * psi(x) * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

1, 3, 4, 7, 8, 4, 9, 8, 4, 16, 9, 8, 20, 8, 8, 11, 8, 12, 20, 20, 8, 15, 16, 12, 20, 16, 8, 24, 21, 8, 20, 8, 16, 28, 24, 8, 17, 32, 12, 36, 16, 8, 24, 16, 24, 19, 20, 20, 32, 16, 12, 28, 16, 20, 44, 27, 12, 36, 24, 16, 28, 24, 16, 28, 32, 12, 25, 32, 12, 48

Offset

0,2

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

Euler transform of period 8 sequence [ 3, -2, 3, -5, 3, -2, 3, -3, ...].

Example

1 + 3*x + 4*x^2 + 7*x^3 + 8*x^4 + 4*x^5 + 9*x^6 + 8*x^7 + 4*x^8 + ...
q + 3*q^9 + 4*q^17 + 7*q^25 + 8*q^33 + 4*q^41 + 9*q^49 + 8*q^57 + 4*q^65 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/2 EllipticTheta[ 2, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^4], {q, 0, 2 n + 1/4}]; Table[a[n], {n, 0, 80}]

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^4 + A)^3 / (eta(x + A)^3 * eta(x^8 + A)^2), n))}

Crossrefs

Sequence in context: A054058 A056007 A316973 * A246847 A193406 A104426

Adjacent sequences: A213619 A213620 A213621 * A213623 A213624 A213625

Keyword

nonn

Author

Michael Somos, Jun 16 2012

Status

approved