A143379 Expansion of q^(-7/24) * eta(q) * eta(q^4)^2 / eta(q^2) in powers of q.

1, -1, 0, -1, -1, 1, 1, 1, -1, 1, 0, 1, 0, 0, -2, -1, 0, 0, -1, 1, 1, -2, 0, 0, 0, 1, 1, 0, 2, 0, 1, -1, -1, 0, 1, -1, 0, 0, 1, 0, -1, -1, 0, -1, -1, -1, 0, 0, 0, 1, 0, 1, 0, 1, -1, -1, 2, 0, -1, 1, -1, 1, 0, 3, 1, -1, 0, 0, 0, 1, -2, 0, 0, -1, -1, 0, -1, 0, 1, 0, 0, 1, -1, -1, -1, 0, 0, 0, 0, -1, 0, -2, 0, 1, 2, 1, -1, 0, 2, 1, 0, 0, 0, 0, 1

Offset

0,15

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 psi(x^2) * f(-x) = psi(-x) * f(-x^4) = chi(-x) * f(-x^4)^2 = psi(-x)^2 / chi(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Apr 07 2015

Euler transform of period 4 sequence [ -1, 0, -1, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 72^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A143377.

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

Convolution of A000009 and A134343. - Michael Somos, Jul 11 2012

-2 * a(n) = A143377(4*n + 1). 2 * a(n) = A143380(4*n + 1).

a(2*n) = A214302(n). a(2*n + 1) = - A214303(n). - Michael Somos, Jul 11 2012

Example

G.f. = 1 - x - x^3 - x^4 + x^5 + x^6 + x^7 - x^8 + x^9 + x^11 - 2*x^14 - x^15 - x^18 + ...
G.f. = q^7 - q^31 - q^79 - q^103 + q^127 + q^151 + q^175 - q^199 + q^223 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^4]^2 / QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Jul 11 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^4]^2, {x, 0, n}]; (* Michael Somos, Apr 07 2015 *)

Prog

(PARI) {a(n) = my(A, p, e, x); if( n<0, 0, n = n*4 + 1; A = factor(6*n + 1); simplify( I^n / -2 * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, p%8==5 || p%24==23, !(e%2), p%8==3 || p%24==17, (-1)^(e\2)*!(e%2), for(i=1, sqrtint(p\6), if( issquare(p - 6*i^2, &x), break)); (e+1) * (kronecker(12, x) * I^((p-1) / 6))^e))))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A), n))};

Crossrefs

Cf. A000009, A134343, A143377, A143380.

Sequence in context: A225245 A204770 A333382 * A254605 A335664 A269518

Adjacent sequences: A143376 A143377 A143378 * A143380 A143381 A143382

Keyword

sign

Author

Michael Somos, Aug 11 2008

Status

approved