A224216 Expansion of q * f(-q,-q^7)^2 / (phi(q^2) * psi(-q)) in powers of q where phi(), psi(), f(,) are Ramanujan theta functions.

1, -1, -2, 3, 4, -6, -8, 11, 15, -20, -26, 34, 44, -56, -72, 91, 114, -143, -178, 220, 272, -334, -408, 498, 605, -732, -884, 1064, 1276, -1528, -1824, 2171, 2580, -3058, -3616, 4269, 5028, -5910, -6936, 8124, 9498, -11088, -12922, 15034, 17468, -20264

Offset

1,3

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 = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q * (f(-q^8) / f(q^2))^2 * f(-q,-q^7) / f(-q^3,-q^5) = q * f(-q,-q^7) * f(-q^2,-q^6)^2 / (f(-q^3,-q^5) * f(-q^4,-q^4)^2) in powers of q where f() is Ramanujan's theta function.

Euler transform of period 8 sequence [ -1, -2, 1, 4, 1, -2, -1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v) * (1 + 2*u) - (u + u^2) * (1 - 4*v - 4*v^2).

a(n) = (-1)^floor(n \ 2) * A115671(n) unless n=0.

Example

q - q^2 - 2*q^3 + 3*q^4 + 4*q^5 - 6*q^6 - 8*q^7 + 11*q^8 + 15*q^9 + ...

Mathematica

a[ n_] := (-1)^Floor[ n / 2] SeriesCoefficient[ (QPochhammer[ -q] / QPochhammer[ q] - 1) / 2, {q, 0, n}]
a[ n_] := SeriesCoefficient[ q^(1/2) QPochhammer[ -q] EllipticTheta[ 2, 0, q^2] / EllipticTheta[ 4, 0, q^4]^2 QPochhammer[ q, q^8]^2 QPochhammer[ q^7, q^8]^2 / 2, {q, 0, n}]

Prog

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

Crossrefs

Cf. A115671, A208856.

Sequence in context: A175864 A133153 A100673 * A245432 A115671 A208856

Adjacent sequences: A224213 A224214 A224215 * A224217 A224218 A224219

Keyword

sign

Author

Michael Somos, Apr 01 2013

Status

approved