A232635 Expansion of psi(x) * phi(x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

1, 1, 4, 5, 4, 8, 1, 4, 8, 4, 13, 12, 4, 8, 8, 1, 12, 8, 8, 12, 16, 13, 4, 16, 4, 12, 20, 8, 5, 12, 12, 12, 16, 8, 8, 20, 17, 16, 16, 8, 20, 24, 8, 8, 16, 1, 24, 16, 8, 12, 20, 16, 12, 28, 12, 33, 20, 8, 16, 12, 16, 20, 24, 8, 16, 32, 1, 12, 32, 8, 16, 32, 8

Offset

0,3

Comments

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

Theta function of cubic lattice with respect to point (1/4, 0, 0).

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^4)^10 / (eta(q) * eta(q^2)^2 * eta(q^8)^4) in powers of q.

Euler transform of period 8 sequence [ 1, 3, 1, -7, 1, 3, 1, -3, ...].

a(3*n + 2) = 4 * A213023(n).

Example

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

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q^4]^10 / (QPochhammer[ q] QPochhammer[ q^2]^2 QPochhammer[ q^8]^4), {q, 0, n}]

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^10 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^8 + A)^4), n))}

Crossrefs

Cf. A213023.

Sequence in context: A248671 A378211 A343442 * A201296 A350089 A246954

Adjacent sequences: A232632 A232633 A232634 * A232636 A232637 A232638

Keyword

nonn

Author

Michael Somos, Nov 27 2013

Status

approved