A225853 Expansion of phi(x) / f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions.

1, 2, 0, 0, 3, 2, 0, 0, 4, 6, 0, 0, 7, 8, 0, 0, 13, 14, 0, 0, 19, 20, 0, 0, 29, 34, 0, 0, 43, 46, 0, 0, 62, 70, 0, 0, 90, 96, 0, 0, 126, 138, 0, 0, 174, 186, 0, 0, 239, 262, 0, 0, 325, 346, 0, 0, 435, 472, 0, 0, 580, 620, 0, 0, 769, 826, 0, 0, 1007, 1072, 0

Offset: 0

Author

Michael Somos, May 17 2013

Keywords

• nonn

Comments

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

Examples

			1 + 2*x + 3*x^4 + 2*x^5 + 4*x^8 + 6*x^9 + 7*x^12 + 8*x^13 + 13*x^16 + ...
1/q + 2*q^5 + 3*q^23 + 2*q^29 + 4*q^47 + 6*q^53 + 7*q^71 + 8*q^77 + 13*q^95 + ...
		

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

Crossrefs

Cf. A029552, A143161.

Programs

Mathematica

a[n_]:= SeriesCoefficient[EllipticTheta[3,0,q]/QPochhammer[q^4],{q,0,n}];
a[n_]:= SeriesCoefficient[QPochhammer[q^2,q^4]^3/QPochhammer[q,q^2]^2, {q,0,n}];

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^3), n))}

Formula

Expansion of chi(x)^2 * chi(-x^2) = chi(x)^3 * chi(-x) = chi(-x^2)^3 / chi(-x)^2 in powers of x where chi() is a Ramanujan theta function.

Expansion of q^(1/4) * eta(q^2)^5 / (eta(q)^2 * eta(q^4)^3) in powers of q.

Euler transform of period 4 sequence [ 2, -3, 2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A029552.

G.f.: Product_{k>0} (1 - x^(4*k-2))^3 / (1 - x^(2*k-1))^2 = (Sum_{k in Z} x^k^2) / (Product_{k>0} (1 - x^(4*k))).

a(n) = (-1)^n * A143161(n). a(4*n + 2) = a(4*n + 3) = 0.