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

Author

Michael Somos, Sep 04 2014

Keywords

• nonn

Comments

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

Examples

			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 + ...
		

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

Crossrefs

Cf. A246816.

Programs

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 *)

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))};

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.