A138514 Expansion of q^(-1/8) * eta(q^2)^4 / (eta(q) * eta(q^4)) in powers of q.

1, 1, -2, -1, 0, -2, 1, 0, 0, 2, 1, 2, -2, 0, 2, 1, 0, -2, 0, -2, 0, -1, 0, 0, -2, 0, 0, 0, -1, 2, -2, 0, 2, 0, 0, 2, 3, 0, 0, -2, 0, 0, 2, 0, 2, 1, -2, 0, 0, 0, -2, -2, 0, 2, -2, 1, -2, -2, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, -2, 2, 0, 2, 2, 0, 2, 1, 0, -2, 0, 2, 0, -2, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, -2, -2, 0, 0, 0, 2, -2, 0, 0, -2

Offset

0,3

Comments

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

Number 70 of the 74 eta-quotients listed in Table I of Martin (1996).

A030204, A083650 and A138514 are the same except for signs. - N. J. A. Sloane, May 07 2010

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(x) * f(-x^2) = psi(-x) * phi(x) = chi(x) * f(-x^2)^2 = psi(x) * phi(-x^2) = f(x)^2 / chi(x) = f(x)^3 / phi(x) = f(-x^2)^3 / psi(-x) = phi(x)^2 / chi(x)^3 = chi(x)^3 * psi(-x)^2 = (f(x)^3 * psi(-x))^(1/2) = (f(-x^2)^3 * phi(x))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.

Expansion of psi(i * x) * psi(-i * x) in powers x^2 where i^2 = -1 and psi() is a Ramanujan theta function. - Michael Somos, Feb 16 2014

Euler transform of period 4 sequence [ 1, -3, 1, -2, ...].

a(n) = b(8*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = 0 if p == 3, 5, 7 (mod 8) and e odd, b(p^e) = 1 if p == 3 (mod 4) and e even, b(p^e) = (-1)^(e/2) if p == 5 (mod 8) and e even, b(p^e) = e+1 if p == 1 (mod 8) and p = x^2 + 64*y^2, b(p^e) = (-1)^e * (e+1) if p == 1 (mod 8) and p is not of the form x^2 + 64*y^2.

a(9*n + 1) = a(n), a(9*n + 4) = a(9*n + 7) = 0. a(n) = (-1)^n * A030204(n) = (-1)^floor((n+1)/2) * A083650(n).

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 16 (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 10 2015

Example

G.f. = 1 + x - 2*x^2 - x^3 - 2*x^5 + x^6 + 2*x^9 + x^10 + 2*x^11 - 2*x^12 + ...
G.f. = q + q^9 - 2*q^17 - q^25 - 2*q^41 + q^49 + 2*q^73 + q^81 + 2*q^89 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ -x], {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^4 / (QPochhammer[ x] QPochhammer[ x^4]), {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 / (eta(x + A) * eta(x^4 + A)), n))};
(PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 8*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p%8==1, (e+1) * if( qfbclassno( -8 * p) / 4 % 2, (-1)^e, 1), if( e%2==0, (-1)^(e/2 * (p%8==5)))))) };
(PARI) {a(n) = if( n<0, 0, n = 8*n + 1; (qfrep([ 1, 0; 0, 64], n) - qfrep([ 4, 2; 2, 17], n))[n])};

Crossrefs

Cf. A030204, A083650.

Sequence in context: A190893 A030204 A083650 * A286137 A320658 A284966

Adjacent sequences: A138511 A138512 A138513 * A138515 A138516 A138517

Keyword

sign

Author

Michael Somos, Mar 22 2008

Status

approved