A244600 Expansion of f(-x) / f(-x^7) in powers of x where f() is a Ramanujan theta function.

1, -1, -1, 0, 0, 1, 0, 2, -1, -1, 0, 0, 0, 0, 3, -3, -2, 0, 0, 1, 0, 5, -3, -3, 0, 0, 2, 0, 8, -6, -5, 0, 0, 3, 0, 11, -8, -7, 0, 0, 3, 0, 17, -13, -11, 0, 0, 6, 0, 24, -17, -14, 0, 0, 7, 0, 34, -25, -21, 0, 0, 11, 0, 47, -33, -28, 0, 0, 14, 0, 64, -47, -39

Offset

0,8

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(1/4) * eta(q) / eta(q^7) in powers of q.

Euler transform of period 7 sequence [ -1, -1, -1, -1, -1, -1, 0, ...].

Given g.f. A(x), then B(q) = A(q^4) / q satisifes 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (2 + u*v).

G.f. is a period 1 Fourier series which satisfies f(-1 / (112 t)) = 7^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A035985.

G.f.: Product_{k>0} (1 - x^k) / (1 - x^(7*k)).

Convolution inverse of A035985.

a(7*n + 3) = a(7*n + 4) = a(7*n + 6) = 0.

a(n) = -(1/n)*Sum_{k=1..n} A113957(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

Example

G.f. = 1 - x - x^2 + x^5 + 2*x^7 - x^8 - x^9 + 3*x^14 - 3*x^15 - 2*x^16 + ...
G.f. = q^-1 - q^3 - q^7 + q^19 + 2*q^27 - q^31 - q^35 + 3*q^55 - 3*q^59 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] / QPochhammer[ x^7], {x, 0, n}];

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^7 + A), n))};

Crossrefs

Cf. A035985.

Sequence in context: A035185 A339659 A338971 * A288558 A362831 A269241

Adjacent sequences: A244597 A244598 A244599 * A244601 A244602 A244603

Keyword

sign

Author

Michael Somos, Jul 01 2014

Status

approved