A137676 Expansion of f(-x^2, -x^3) / f(-x, -x^3) in powers of x where f(, ) is Ramanujan's general theta function.

1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 3, 3, 0, 0, 4, 4, 0, 0, 5, 6, 0, 0, 7, 7, 0, 0, 9, 10, 0, 0, 12, 12, 0, 0, 15, 16, 0, 0, 19, 20, 0, 0, 24, 26, 0, 0, 30, 31, 0, 0, 37, 40, 0, 0, 46, 48, 0, 0, 57, 60, 0, 0, 69, 72, 0, 0, 84, 89, 0, 0, 102, 106, 0

Offset: 0

Author

Michael Somos, Feb 04 2008

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x + x^4 + x^5 + x^8 + 2*x^9 + 2*x^12 + 2*x^13 + 3*x^16 + 3*x^17 + ...
G.f. = 1/q + q^9 + q^39 + q^49 + q^79 + 2*q^89 + 2*q^119 + 2*q^129 + 3*q^159 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 36, Eq. (4.11). MR0858826 (88b:11063).
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A122129, A122135.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] / (QPochhammer[ x^4] QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}]; (* Michael Somos, Oct 08 2015 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2) / QPochhammer[ x^4, x^4, k], {k, 0, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Oct 08 2015 *)
a[ n_] := SeriesCoefficient[ Sqrt[2] x^(1/8) QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5] QPochhammer[ x^5] / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}]; (* Michael Somos, Oct 08 2015 *)

PARI

{a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n), x^k^2 / prod(i=1, k, 1 - x^(4*i), 1 + x * O(x^(n - k^2)))), n))};

Formula

Expansion of f(-x^2) * f(-x^5) / (f(-x^4) * f(-x, -x^4)) in powers of x where f(, ) is Ramanujan's general theta function.

Expansion of (f(-x^13, -x^17) + x * f(-x^7, -x^23)) / f(-x^4) in powers of x where f(, ) is Ramanujan's general theta function.

Euler transform of period 20 sequence [ 1, -1, 0, 1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 1, 0, -1, 1, 0, ...].

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

a(4*n) = A122129(n). a(4*n + 1) = A122135(n). a(4*n + 2) = a(4*n + 3) = 0.

G.f.: (Sum_{k in Z} (-1)^k^2 * x^(k * (5*k + 1) / 2)) / (Sum_{k in Z} (-1)^k^2 * x^(k * (2*k + 1))). - Michael Somos, Oct 08 2015