A230322 Expansion of f(-x^3, -x^4) / f(-x^2,-x^5) in powers of x where f(,) is Ramanujan's two-variable theta function.

1, 0, 1, -1, 0, 0, 0, 1, -1, 1, -1, 0, 0, -1, 2, -1, 2, -2, 0, 0, -1, 3, -3, 3, -3, 1, 0, -2, 5, -4, 5, -5, 1, 0, -3, 7, -7, 8, -7, 2, 0, -5, 11, -10, 12, -11, 3, 1, -7, 15, -16, 17, -15, 5, 1, -11, 22, -22, 25, -22, 7, 2, -15, 31, -33, 35, -30, 11, 2, -22

Offset

0,15

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

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

G.f.: Product_{k>0} (1 - x^(7*k - 3)) * (1 - x^(7*k - 4)) / ((1 - x^(7*k - 2)) * (1 - x^(7*k - 5))).

- a(n) = A229894(7*n + 1).

G.f.: B(x) / C(x), where B(x) is the g.f. of A375150 and C(x) is the g.f. of A375107. - Seiichi Manyama, Aug 03 2024

Example

G.f. = 1 + x^2 - x^3 + x^7 - x^8 + x^9 - x^10 - x^13 + 2*x^14 - x^15 + ...
G.f. = 1/q + q^13 - q^20 + q^48 - q^55 + q^62 - q^69 - q^90 + 2*q^97 - ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^7] QPochhammer[ x^4, x^7] / (QPochhammer[ x^2, x^7] QPochhammer[ x^5, x^7]), {x, 0, n}]

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^[ 0, 0, -1, 1, 1, -1, 0][k%7+1], 1 + x * O(x^n)), n))}

Crossrefs

Cf. A229894, A375107, A375150.

Sequence in context: A376596 A088151 A286129 * A174950 A159906 A263444

Adjacent sequences: A230319 A230320 A230321 * A230323 A230324 A230325

Keyword

sign

Author

Michael Somos, Oct 16 2013

Status

approved