A246811 Expansion of phi(x)^2 * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

1, 4, 4, 0, 5, 12, 4, 0, 8, 12, 8, 0, 5, 16, 12, 0, 8, 24, 4, 0, 16, 12, 12, 0, 9, 24, 12, 0, 8, 36, 12, 0, 16, 12, 16, 0, 8, 28, 16, 0, 17, 36, 8, 0, 24, 24, 8, 0, 8, 36, 28, 0, 16, 36, 12, 0, 16, 24, 20, 0, 13, 24, 24, 0, 24, 60, 8, 0, 16, 36, 16, 0, 16, 28

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of psi(x)^4 / phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

Expansion of q^(-1/2) * eta(q^2)^10 * eta(q^8)^2 / (eta(q)^4 * eta(q^4)^5) in powers of q.

Euler transform of period 8 sequence [4, -6, 4, -1, 4, -6, 4, -3, ...].

2 * a(n) = A033717(4*n + 2). a(2*n) = A045834(n). a(4*n) = A213022(n). a(4*n + 1) = 4 * A008443(n). a(4*n + 2) = 4 * A045831(n). a(4*n + 3) = 0.

Example

G.f. = 1 + 4*x + 4*x^2 + 5*x^4 + 12*x^5 + 4*x^6 + 8*x^8 + 12*x^9 + ...
G.f. = q + 4*q^3 + 4*q^5 + 5*q^9 + 12*q^11 + 4*q^13 + 8*q^17 + 12*q^19 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^2 EllipticTheta[ 2, 0, x^2] / (2 x^(1/2)), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)]^4 / (16 x^(1/2) EllipticTheta[ 3, 0, x^2]), {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^8 + A)^2 / (eta(x + A)^4 * eta(x^4 + A)^5), n))};

Crossrefs

Cf. A008443, A033717, A045831, A045834, A213022.

Sequence in context: A159796 A069523 A021231 * A197140 A155502 A228048

Adjacent sequences: A246808 A246809 A246810 * A246812 A246813 A246814

Keyword

nonn

Author

Michael Somos, Sep 03 2014

Status

approved