A275372 Expansion of f(-x) * f(-x^2)^4 / phi(x^2) in powers of x where phi(), f() are Ramanujan theta functions.

1, -1, -7, 6, 20, -13, -34, 15, 53, -25, -91, 52, 135, -65, -180, 82, 253, -133, -343, 160, 449, -207, -603, 306, 780, -348, -979, 438, 1241, -600, -1557, 703, 1924, -890, -2375, 1115, 2910, -1300, -3535, 1620, 4318, -1993, -5198, 2335, 6180, -2783, -7420

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

Expansion of q^(-3/8) * eta(q) * eta(q^2)^6 * eta(q^8)^2 / eta(q^4)^5 in powers of q.

Euler transform of period 8 sequence [ -1, -7, -1, -2, -1, -7, -1, -4, ...].

G.f.: Product_{k>0} (1 - x^k)^4 * (1 + x^k)^3 * (1 + x^(4*k))^2 / (1 + x^(2*k))^3.

2 * a(n) = - A279955(2*n + 1).

Example

G.f. = 1 - x - 7*x^2 + 6*x^3 + 20*x^4 - 13*x^5 - 34*x^6 + 15*x^7 + 53*x^8 + ...
G.f. = q^3 - q^11 - 7*q^19 + 6*q^27 + 20*q^35 - 13*q^43 - 34*q^51 + 15*q^59 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2]^4 / EllipticTheta[ 3, 0, x^2], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^7 / (QPochhammer[ -x] QPochhammer[ -x^2]^2), {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)^6 * eta(x^8 + A)^2 / eta(x^4 + A)^5, n))};

Crossrefs

Cf. A279955.

Sequence in context: A298377 A299244 A249114 * A338137 A341289 A322049

Adjacent sequences: A275369 A275370 A275371 * A275373 A275374 A275375

Keyword

sign

Author

Michael Somos, Dec 25 2016

Status

approved