A263548 Expansion of f(x, x) * f(x^2, x^10) in powers of x where f(, ) is Ramanujan's general theta function.

1, 2, 1, 2, 2, 0, 2, 0, 0, 2, 1, 4, 0, 0, 2, 0, 3, 2, 2, 2, 2, 0, 0, 0, 0, 4, 2, 2, 0, 0, 0, 0, 3, 2, 0, 2, 4, 0, 2, 0, 0, 4, 1, 2, 0, 0, 4, 0, 2, 2, 0, 4, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 2, 2, 3, 4, 2, 0, 2, 0, 0, 0, 2, 2, 0, 0, 2, 0, 3, 6, 2, 2, 2, 0, 0

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 phi(x) * chi(x^2) * psi(-x^6) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Expansion of q^(-2/3) * eta(q^2)^4 * eta(q^6) * eta(q^24) / (eta(q)^2 * eta(q^8) * eta(q^12)) in powers of q.

Euler transform of period 24 sequence [ 2, -2, 2, -2, 2, -3, 2, -1, 2, -2, 2, -2, 2, -2, 2, -1, 2, -3, 2, -2, 2, -2, 2, -2, ...].

a(n) = A000377(3*n + 2) = A192013(3*n + 2) = A115660(6*n + 4).

Example

G.f. = 1 + 2*x + x^2 + 2*x^3 + 2*x^4 + 2*x^6 + 2*x^9 + x^10 + 4*x^11 + ...
G.f. = q^2 + 2*q^5 + q^8 + 2*q^11 + 2*q^14 + 2*q^20 + 2*q^29 + q^32 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^2, x^12] QPochhammer[ -x^10, x^12] QPochhammer[ x^12], {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^2, x^4] EllipticTheta[ 2, Pi/4, x^3] / (2^(1/2) x^(3/4)), {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A) * eta(x^24 + A) / (eta(x + A)^2 * eta(x^8 + A) * eta(x^12 + A)), n))};

Crossrefs

Cf. A000377, A115660, A192013.

Sequence in context: A035146 A035216 A258587 * A058496 A137579 A337108

Adjacent sequences: A263545 A263546 A263547 * A263549 A263550 A263551

Keyword

nonn

Author

Michael Somos, Oct 20 2015

Status

approved