A224833 Expansion of phi(-x)^2 * chi(-x) * psi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

1, -5, 8, -4, 4, -13, 12, -4, 5, -16, 24, -8, 4, -20, 12, -8, 9, -20, 32, -4, 12, -29, 12, -8, 8, -36, 40, -8, 8, -20, 24, -16, 8, -25, 40, -12, 12, -32, 24, -12, 13, -48, 40, -8, 8, -40, 36, -8, 16, -20, 56, -16, 12, -52, 12, -20, 13, -36, 56, -16, 20, -40, 24

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^(-1/3) * eta(q)^5 * eta(q^6)^2 / (eta(q^2)^3 * eta(q^3)) in powers of q.

Euler transform of period 6 sequence [ -5, -2, -4, -2, -5, -3, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 73728^(1/2) (t / i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227595.

-2 * a(n) = A224822(3*n + 1).

Example

1 - 5*x + 8*x^2 - 4*x^3 + 4*x^4 - 13*x^5 + 12*x^6 - 4*x^7 + 5*x^8 - 16*x^9 + ...
q - 5*q^4 + 8*q^7 - 4*q^10 + 4*q^13 - 13*q^16 + 12*q^19 - 4*q^22 + 5*q^25 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-1/3)* eta[q]^5*eta[q^6]^2/(eta[q^2]^3*eta[q^3]), {q, 0, n}];  Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 19 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^5 * eta(x^6 + A)^2 / (eta(x^2 + A)^3 * eta(x^3 + A)), n))}

Crossrefs

Cf. A224822, A227595.

Sequence in context: A199450 A366072 A019649 * A246926 A199288 A099878

Adjacent sequences: A224830 A224831 A224832 * A224834 A224835 A224836

Keyword

sign

Author

Michael Somos, Jul 21 2013

Status

approved