A279918 Expansion of f(-x^2)^7 / (f(x) * f(-x^8)^2) in powers of x where f() is a Ramanujan theta function.

1, -1, -5, 4, 5, 0, 11, -15, -18, 3, -10, 29, 10, 11, 37, -51, -16, -30, -65, 62, 53, 22, 50, -61, -52, -4, -81, 120, 62, 0, 124, -182, -85, -43, -157, 171, 123, 60, 202, -198, -174, 0, -190, 301, 117, 54, 278, -375, -171, -153, -399, 370, 300, 108, 408, -451

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

Expansion of phi(-x^2)^4 * chi(x^2)^2 / chi(x) in powers of x where chi(), phi() are Ramanujan theta functions.

Expansion of q^(-1/8) * eta(q) * eta(q^2)^4 * eta(q^4) / eta(q^8)^2 in powers of q.

Euler transform of period 8 sequence [ -1, -5, -1, -6, -1, -5, -1, -4, ...].

a(n) = A279955(2*n).

Example

G.f. = 1 - x - 5*x^2 + 4*x^3 + 5*x^4 + 11*x^6 - 15*x^7 - 18*x^8 + ...
G.f. = q^-1 - q^7 - 5*q^15 + 4*q^23 + 5*q^31 + 11*q^47 - 15*q^55 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^3 QPochhammer[ -x^2, x^4]^2 QPochhammer[ x]^4, {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2]^4 QPochhammer[ -x^2, x^4]^2 QPochhammer[ x, -x], {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A)^4 * eta(x^4 + A) / eta(x^8 + A)^2, n))};
(PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q)*eta(q^2)^4*eta(q^4)/eta(q^8)^2)} \\ Altug Alkan, Mar 21 2018

Crossrefs

Cf. A279955.

Sequence in context: A291069 A019117 A204372 * A273986 A246729 A337029

Adjacent sequences: A279915 A279916 A279917 * A279919 A279920 A279921

Keyword

sign

Author

Michael Somos, Dec 23 2016

Status

approved