A128765 Expansion of psi(q) * psi(q^10) / ( psi(q^2) * psi(q^5)) in powers of q where psi() is a Ramanujan theta function.

1, 1, -1, 0, 1, -1, -2, 0, 2, 0, 0, 2, 1, -1, -2, -2, 0, 1, -2, 0, 6, 3, -5, -2, 5, -3, -10, 2, 8, -1, -2, 6, 6, -2, -8, -6, 2, 2, -8, -2, 21, 11, -18, -4, 18, -11, -32, 4, 26, -1, -10, 18, 20, -8, -26, -18, 10, 8, -26, -2, 61, 27, -53, -12, 52, -26, -88, 12, 74, -6, -32, 42, 58, -17, -74, -40, 34, 16, -74, -8, 156, 66, -136, -26

Offset

0,7

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^(-1/2) * eta(q^2)^3 * eta(q^5) * eta(q^20)^2 / (eta(q) * eta(q^4)^2 * eta(q^10)^3) in powers of q.

Euler transform of period 20 sequence [ 1, -2, 1, 0, 0, -2, 1, 0, 1, 0, 1, 0, 1, -2, 0, 0, 1, -2, 1, 0, ...].

Given g.f. A(x), then B(q) = q*A(q^2) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (u - v)^2.

G.f.: Product_{k>0} (1 + x^k) * (1 + x^(10*k))^2 / ( (1 + x^(2*k))^2* (1+x^(5*k)) ).

Example

G.f. = 1 + x - x^2 + x^4 - x^5 - 2*x^6 +  2*x^8 + 2*x^11 + x^12 - x^13 + ...
G.f. = q + q^3 - q^5 + q^9 - q^11 - 2*q^13 + 2*q^17 + 2*q^23 + q^25 - q^27 + ...

Mathematica

a[ n_] := SeriesCoefficient[ x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^5] / (EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^(5/2)]), {x, 0, n}]; (* Michael Somos, Apr 26 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^5 + A) * eta(x^20 + A)^2 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^10 + A)^3), n))};

Crossrefs

Sequence in context: A221645 A216176 A359007 * A193511 A254218 A263147

Adjacent sequences: A128762 A128763 A128764 * A128766 A128767 A128768

Keyword

sign

Author

Michael Somos, Mar 25 2007

Status

approved