A246650 Expansion of phi(x) * chi(-x) * psi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions

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

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/3) * eta(q^2)^4 * eta(q^6)^2 / (eta(q) * eta(q^3) * eta(q^4)^2) in powers of q.

a(2*n) = A129451(n). a(4*n) = A123884(n). a(4*n + 1) = A122861(n). a(4*n + 2) = -2 * A121361(n). a(4*n + 3) = 0.

a(8*n) = A131961(n). a(8*n + 1) = A097195(n). a(8*n + 2) = -2 * A131962(n). a(8*n + 4) = 2 * A131963(n). a(8*n + 6) = -2 * A131964(n).

a(16*n + 1) = A123884(n). a(16*n + 5) = -3 * A033687(n). a(16*n + 9) = 2 * A121361(n). a(16*n + 13) = 0.

Example

G.f. = 1 + x - 2*x^2 + 2*x^4 - 3*x^5 - 2*x^6 + x^8 + 2*x^9 - 2*x^10 + ...
G.f. = q + q^4 - 2*q^7 + 2*q^13 - 3*q^16 - 2*q^19 + q^25 + 2*q^28 - ...

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^2 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A)^2), n))};

Crossrefs

Cf. A033687, A121361, A122861, A123884, A129451, A131961, A131962, A131963, A131964.

Sequence in context: A079957 A202036 A246752 * A264940 A272800 A104513

Adjacent sequences: A246647 A246648 A246649 * A246651 A246652 A246653

Keyword

sign

Author

Michael Somos, Aug 31 2014

Status

approved