A228834 Expansion of phi(-x^2)^2 * psi(-x)^4 in powers of x where phi(), psi() are Ramanujan theta functions.

1, -4, 2, 8, -7, 4, -14, -8, 18, 12, 32, -40, -21, -8, -14, 32, 16, 16, -30, 56, -14, -28, -14, -16, -15, -72, 66, 8, 48, 52, 82, -56, -28, -4, -160, -56, 66, 84, -32, 16, -95, 140, 36, 56, -30, -112, 128, 24, -14, -28, -94, -152, 64, -156, 18, 120, 98, -80

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 q^(-1/2) * (eta(q)^2 * eta(q^4))^2 in powers of q.

Euler transform of period 4 sequence [ -4, -4, -4, -6, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 1024 (t / i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A034952.

G.f. (Product_{k>0} (1 - x^k)^2 * (1 - x^(4*k)))^2.

a(2*n) = A228831(n). a(2*n + 1) = -4 * A034952(n).

Example

G.f. = 1 - 4*x + 2*x^2 + 8*x^3 - 7*x^4 + 4*x^5 - 14*x^6 - 8*x^7 + 18*x^8 + ...
G.f. = q - 4*q^3 + 2*q^5 + 8*q^7 - 7*q^9 + 4*q^11 - 14*q^13 - 8*q^15 + 18*q^17 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4 * QPochhammer[ x^4]^2, {x, 0 , n}];

Prog

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

Crossrefs

Cf. A034952, A228831.

Sequence in context: A131783 A094312 A182848 * A197016 A198145 A143942

Adjacent sequences: A228831 A228832 A228833 * A228835 A228836 A228837

Keyword

sign,changed

Author

Michael Somos, Sep 04 2013

Status

approved