A256505 Expansion of phi(x^3) * phi(-x^48) / chi(-x^16) in powers of x where phi(), chi() are Ramanujan theta functions.

1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0

Offset

0,4

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 q^(-2/3) * eta(q^6)^5 * eta(q^32) * eta(q^48)^2 / (eta(q^3)^2 * eta(q^12)^2 * eta(q^16) * eta(q^96)) in powers of q.

Euler transform of a period 96 sequence.

a(n) = A257403(3*n + 2) unless n == 5 (mod 8).

a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = a(16*n + 4) = a(16*n + 8) = 0.

a(4*N) = A257399(n). a(8*n+3) = 2*A255318(n). a(16*n) = A257398(n). a(16*n+12) = 2*A255317(n).

Example

G.f. = 1 + 2*x^3 + 2*x^12 + x^16 + 2*x^19 + 2*x^27 + 2*x^28 + x^32 + ...
G.f. = q^2 + 2*q^11 + 2*q^38 + q^50 + 2*q^59 + 2*q^83 + 2*q^86 + q^98 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] EllipticTheta[ 4, 0, x^48] QPochhammer[ -x^16, x^16], {x, 0, n}];

Prog

(PARI) {a(n) = my(A, p, e); if( n<0 || n%8 == 5, 0, A = factor(3*n + 2); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p+e==3, p%8 > 4, 1-e%2, e+1)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^5 * eta(x^32 + A) * eta(x^48 + A)^2 / (eta(x^3 + A)^2 * eta(x^12 + A)^2 * eta(x^16 + A) * eta(x^96 + A)), n))};

Crossrefs

Cf. A255317, A255318, A257398, A257399, A257403.

Sequence in context: A368072 A233441 A255365 * A337196 A073346 A114099

Adjacent sequences: A256502 A256503 A256504 * A256506 A256507 A256508

Keyword

nonn

Author

Michael Somos, Apr 22 2015

Status

approved