login
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.
2
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
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
KEYWORD
sign
AUTHOR
Michael Somos, Oct 16 2013
STATUS
approved