A214639 Expansion of q * f(-q^2, -q^14) / f(-q, q^3) in powers of q where f(,) is Ramanujan's two-variable theta function.

0, 1, 1, 0, -1, -2, -2, 0, 3, 5, 4, 0, -6, -10, -8, 0, 11, 18, 15, 0, -20, -32, -26, 0, 34, 55, 44, 0, -56, -90, -72, 0, 91, 144, 114, 0, -143, -226, -178, 0, 220, 346, 272, 0, -334, -522, -408, 0, 498, 777, 605, 0, -732, -1138, -884, 0, 1064, 1648, 1276, 0

Offset

0,6

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 * f(q, q^7)^2 / (phi(q^4) * psi(q)) in powers of q where phi(), psi() are Ramanujan theta functions.

Euler transform of period 16 sequence [ 1, -1, -1, -1, -1, 1, 1, 2, 1, 1, -1, -1, -1, -1, 1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 2 * u * v * (1 - u - u*v).

a(4*n + 3) = 0. a(4*n) = (-1)^n * A185083(n) unless n=0. a(4*n + 1) = A079006(n). a(4*n + 2) = A210063(n).

a(2*n) = A224216(n).

Example

q + q^2 - q^4 - 2*q^5 - 2*q^6 + 3*q^8 + 5*q^9 + 4*q^10 - 6*q^12 - 10*q^13 + ...

Mathematica

nn = 16*10; b = Flatten[Table[{1, -1, -1, -1, -1, 1, 1, 2, 1, 1, -1, -1, -1, -1, 1, 0}, {nn/16}]]; CoefficientList[x * Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jul 25 2012 *)
a[ n_] :=  SeriesCoefficient[ 2 q^(9/8) QPochhammer[ -q, q^8]^2 QPochhammer[ q^8]^2 QPochhammer[ -q^7, q^8]^2 / (EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 3, 0, q^4]), {q, 0, n}] (* Michael Somos, Apr 02 2013 *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, if( n%4==3, 0, if( n%2==0, n/=2; A = x * O(x^n); (-1)^(n\2) * polcoeff( eta(-x + A) / eta(x + A), n) / 2, n\=4; A = x * O(x^n); (-1)^n * polcoeff( (eta(x^4 + A) / eta(x + A))^2, n))))}
(PARI) {a(n) = if( n<1, 0, n--; polcoeff( prod( k=1, n, (1 -x^k + x * O(x^n))^[ 0, -1, 1, 1, 1, 1, -1, -1, -2, -1, -1, 1, 1, 1, 1, -1][k%16 + 1]), n))}

Crossrefs

Cf. A079006, A185083, A210063, A214263, A224216.

Sequence in context: A283269 A201947 A098816 * A319495 A216973 A061314

Adjacent sequences: A214636 A214637 A214638 * A214640 A214641 A214642

Keyword

sign

Author

Michael Somos, Jul 24 2012

Status

approved