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

0,2

Comments

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

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

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.

Example

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 + ...

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}];

Prog

(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))}

Crossrefs

Cf. A029552, A143161.

Sequence in context: A224777 A259827 A143161 * A342128 A330463 A142886

Adjacent sequences: A225850 A225851 A225852 * A225854 A225855 A225856

Keyword

nonn

Author

Michael Somos, May 17 2013

Status

approved