A260089 Expansion of psi(x^2) * f(x, x^2) in powers of x where psi(), f() are Ramanujan theta functions.

1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 0, 1, 2, 2, 2, 1, 0, 2, 1, 1, 1, 2, 2, 0, 2, 1, 1, 2, 2, 0, 1, 1, 3, 0, 1, 3, 0, 2, 1, 0, 1, 1, 4, 1, 1, 1, 2, 2, 0, 1, 0, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 1, 0, 3, 1, 1, 0, 0, 2, 1, 2, 2, 2, 1, 1, 0, 1, 4, 1, 2, 0, 1, 3, 1, 1, 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 phi(-x^3) * f(-x^4)^2 / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

Expansion of q^(-7/24) * eta(q^3)^2 * eta(q^4)^2 / (eta(q) * eta(q^6)) in powers of q.

Euler transform of period 12 sequence [ 1, 1, -1, -1, 1, 0, 1, -1, -1, 1, 1, -2, ...].

G.f.: Product_{k>0} (1 + x^(2*k)) * (1 - x^(3*k)) * (1 - x^(4*k)) / (1 - x^k + x^(2*k)).

Example

G.f. = 1 + x + 2*x^2 + x^3 + x^4 + x^5 + x^6 + 3*x^7 + x^8 + x^9 + ...
G.f. = q^7 + q^31 + 2*q^55 + q^79 + q^103 + q^127 + q^151 + 3*q^175 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^2 QPochhammer[ x^4]^2 / (QPochhammer[ x] QPochhammer[ x^6]), {x, 0, n}];

Prog

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

Crossrefs

Sequence in context: A293483 A125761 A154950 * A296240 A158464 A056926

Adjacent sequences: A260086 A260087 A260088 * A260090 A260091 A260092

Keyword

nonn

Author

Michael Somos, Jul 15 2015

Status

approved