A246832 Expansion of psi(x) * psi(x^2) * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

1, 1, 3, 4, 2, 5, 2, 3, 7, 5, 5, 6, 5, 3, 10, 6, 3, 7, 7, 4, 11, 9, 3, 14, 8, 8, 5, 4, 7, 10, 13, 7, 8, 10, 7, 15, 8, 4, 17, 10, 8, 11, 10, 5, 16, 11, 4, 11, 15, 8, 14, 10, 7, 22, 8, 9, 14, 8, 13, 21, 12, 5, 13, 15, 6, 21, 15, 9, 13, 8, 11, 9, 12, 15, 20, 21

Offset

0,3

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of psi(x) * psi(x^2)^3 / psi(x^4) in powers of x where psi() is a Ramanujan theta function.

Expansion of q^(-3/8) * eta(q^4)^7 / (eta(q) * eta(q^2) * eta(q^8)^2) in powers of q.

Euler transform of period 8 sequence [1, 2, 1, -5, 1, 2, 1, -3, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 2 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246816.

Example

G.f. = 1 + x + 3*x^2 + 4*x^3 + 2*x^4 + 5*x^5 + 2*x^6 + 3*x^7 + 7*x^8 + ...
G.f. = q^3 + q^11 + 3*q^19 + 4*q^27 + 2*q^35 + 5*q^43 + 2*q^51 + 3*q^59 + ...

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-3/8)* eta[q^4]^7/(eta[q]*eta[q^2]*eta[q^8]^2), {q, 0, 60}], q]]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Aug 05 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^7 / (eta(x + A) * eta(x^2 + A) * eta(x^8 + A)^2), n))};

Crossrefs

Cf. A246816.

Sequence in context: A090131 A152833 A139525 * A133570 A117041 A209688

Adjacent sequences: A246829 A246830 A246831 * A246833 A246834 A246835

Keyword

nonn

Author

Michael Somos, Sep 04 2014

Status

approved