OFFSET
0,1
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^3, b = x^2.
f(a,b) = Sum_{k} a^((k^2+k)/2)*b^((k^2-k)/2) is Ramanujan's two-variable theta function.
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
Eric Weisstein's World of Mathematics, Quintuple Product Identity
FORMULA
Expansion of f(x^4, x^11) - x * f(x, x^14) = f(-x^5, -x^10) * f(-x, -x^4) / f(x^2, x^3) in powers of x where f() is Ramanujan's two-variable theta function.
Expansion of H(x^2) * f(-x) where H() is g.f. of A003106.
Euler transform of period 10 sequence [ -1, -1, -1, 0, -1, 0, -1, -1, -1, -1, ...].
G.f.: Sum_{k} x^(5*k * (3*k + 1) / 2) * (x^(-6*k) - x^(6*k + 2)).
G.f.: Product_{k>0} (1 - x^k) / ((1 - x^(10*k - 4)) * (1 - x^(10*k - 6))).
EXAMPLE
1 - x - x^2 + x^4 + x^11 - x^15 - x^18 + x^23 + x^37 - x^44 - x^49 + ...
q^49 - q^169 - q^289 + q^529 + q^1369 - q^1849 - q^2209 + q^2809 + q^4489 + ...
MATHEMATICA
f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A113431[n_] := SeriesCoefficient[f[-x, -x^2]*f[-x^10, -x^20]/f[-x^4, -x^6], {x, 0, n}]; Table[A113431[n], {n, 0, 50}] (* G. C. Greubel, Jun 17 2017 *)
PROG
(PARI) {a(n) = local(m); if( n<0 || !issquare( n*120 + 49, &m) || 1! = gcd(m, 30), 0, (-1)^( m%30 \ 10))}
(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 - x^k*[ 1, 1, 1, 1, 0, 1, 0, 1, 1, 1][k%10 + 1], 1 + x * O(x^n)), n))}
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 31 2005
STATUS
approved