|
| |
|
|
A092869
|
|
Series expansion of the reciprocal of the Goellnitz-Gordon continued fraction.
|
|
3
| |
|
|
1, -1, 0, 1, -1, 1, 0, -2, 2, -1, 0, 2, -3, 2, 0, -2, 4, -4, 0, 4, -6, 5, 0, -6, 9, -6, 0, 7, -12, 9, 0, -10, 16, -13, 0, 15, -22, 17, 0, -20, 29, -21, 0, 25, -38, 28, 0, -32, 50, -39, 0, 43, -64, 49, 0, -56, 82, -60, 0, 69, -105, 78, 0, -86, 132, -101, 0, 112, -166, 125, 0, -142, 208, -153, 0, 172, -258, 192, 0
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,8
|
|
|
COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
|
|
|
REFERENCES
| S.-D. Chen and S.-S. Huang, On the series expansion of the Goellnitz-Gordon continued fraction, Internat. J. Number Theory, 1 (2005), 53-63.
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc., 42 (2005), 137-162; see Eqs. (9.2),(9.4).
|
|
|
LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
|
|
|
FORMULA
| Expansion of f(-x, -x^7) / f(-x^3, -x^5) in powers of x where f(,) is Ramanujan's two variable theta function. - Michael Somos, Aug 02 2011
Expansion of (phi(x) - phi(x^2)) / (2 * x * psi(x^4)) = 2 * psi(x^4) / (phi(x) + phi(x^2)) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos Feb 15 2006
Euler transform of period 8 sequence [ -1, 0, 1, 0, 1, 0, -1, 0, ...].
G.f. A(x) satisfies both A(-x) * A(x) = A(x^2) and x * A(x)^2 = B(x * A(x^2)) where B(x) = x * (1 - x) / (1 + x).
Given g.f. A(x), then B(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 - v + v^2 + v*u^2.
Given g.f. A(x), then B(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (1 - u*v) * (u + v)^3 - v * (1 + v^2) * (1 - u^4). - Michael Somos Feb 15 2006
Given g.f. A(x), then B(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^5)) where f(u, v) = (u - v) * (1 + u*v)^5 - u * (1-u^4) * (1 + v^2) * (1 - 6*v^2 + v^4). - Michael Somos Feb 15 2006
G.f.: Product_{k>=0} (1 - x^(8*k + 1)) * (1 - x^(8*k + 7)) / ((1 - x^(8*k + 3)) * (1 - x^(8*k + 5))).
Continued fraction 1/(1 + x + x^2/(1 + x^3 + x^4/(1 + x^5 + q^x/(1 + x^7 + ...)))). Convolution inverse of A111374.
a(4*n) = A083365(n). a(4*n + 2) = 0.
|
|
|
EXAMPLE
| 1 - x + x^3 - x^4 + x^5 - 2*x^7 + 2*x^8 - x^9 + 2*x^11 - 3*x^12 + ...
q - q^3 + q^7 - q^9 + q^11 - 2*q^15 + 2*q^17 - q^19 + 2*q^23 + ...
|
|
|
MATHEMATICA
| a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^8] QPochhammer[ x^7, x^8] /(QPochhammer[ x^3, x^8] QPochhammer[ x^5, x^8]), {x, 0, n}] (* Michael Somos, Aug 02 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^2] / (EllipticTheta[ 3, 0, x] + EllipticTheta[ 3, 0, x^2]), {x, 0, n + 1/2}] (* Michael Somos, Aug 02 2011 *)
|
|
|
PROG
| (PARI) {a(n) = local(A, u, v); if( n<0, 0, n = 2*n + 1; A = x; forstep( k=3, n, 2, u = A + x * O(x^k); v = subst(u, x, x^2); A -= x^k * polcoeff(u^2 - v + v*u^2 + v^2, k+1) / 2); polcoeff(A, n))}
(PARI) {a(n) = local(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = x * subst(A, x, x^2); A = sqrt(A * (1 - A) / (1 + A) / x)); polcoeff(A, n))}
(PARI) {a(n) = local(A, A2); if( n<0, 0, A = eta(x^8 + x * O(x^n))^2 / eta(x^4 + x * O(x^n)); A2 = sum( k=1, sqrtint(n), x^k^2 + x^(2*k^2), 1 + x * O(x^n)); polcoeff(A / A2, n))}
(PARI) {a(n) = local(A, A2); if( n<0, 0, A = x * O(x^n); A = eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3; A2 = subst(A, x, x^2); polcoeff( 2 * A^2 * A2^2 / (A^2 + A2), n))}
|
|
|
CROSSREFS
| Cf. A003823, A083365, A091188, A111374.
Sequence in context: A143667 A084934 A125927 * A029337 A060086 A177975
Adjacent sequences: A092866 A092867 A092868 * A092870 A092871 A092872
|
|
|
KEYWORD
| sign
|
|
|
AUTHOR
| Michael Somos, Mar 07 2004; corrected Jun 09 2004
|
| |
|
|
