A030204 Expansion of q^(-1/8) * eta(q) * eta(q^2) 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

Offset

0,3

Comments

Number 66 of the 74 eta-quotients listed in Table I of Martin (1996).

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

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

S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.

M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.

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

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334. [a(n) = coefficient of q^(9m+1) in the q-expansion of the unique normalized newform g of weight 1, level 128, and character chi_2. - N. J. A. Sloane, Oct 18 2014]

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

G.f.: (Sum_{k>0} x^((k^2 - k)/2)) * (Sum_{k in Z} (-1)^k * x^k^2). - Michael Somos, Sep 02 2006

Expansion of psi(x) * phi(-x) = f(-x^2) * f(-x) = f(-x)^2 / chi(-x) = f(-x)^3 / phi(-x) = f(-x^2)^2 * chi(-x) = f(-x^2)^3 / psi(x) = psi(-x) * phi(-x^2) = psi(x)^2 * chi(-x)^3 = phi(-x)^2 / chi(-x)^3 = (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. - Michael Somos, Mar 22 2008

Expansion of psi(x) * psi(-x) in powers of x^2 where psi() is a Ramanujan theta function. - Michael Somos, Oct 11 2013

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

a(3*n) = A107063(n). a(3*n + 2) = -2 * A107064(n). - Michael Somos, Oct 11 2013

a(9*n + 1) = -a(n), a(9*n + 4) = a(9*n + 7) = 0. - Michael Somos, Mar 17 2004

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)^(e/2) if p == 3 (mod 8) and e even, b(p^e) = 1 if p == 5,7 (mod 8) and e even, b(p^e) = e + 1 if p == 1 (mod 8) and p = x^2 + 32*y^2, b(p^e) = (-1)^e * (e + 1) if p == 1 (mod 8) and p is not of the form x^2 + 32*y^2.

a(n) = (-1)^n * A138514(n). Convolution inverse is A002513.

G.f.: exp(Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

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 - 2*q^97 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 / QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, I x] / (4 Sqrt[ x] I^(1/4)), {x, 0, 4 n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, Pi/4, x] / (2^(3/2) x^(1/2)), {x, 0, 4 n}]; (* Michael Somos, Jan 31 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + 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(-4*p)%8, (-1)^e, 1), e%2==0, (-1)^(e/2*(p%8<5)))))}; /* Michael Somos, Jul 26 2006 */
(PARI) {a(n) = if( n<0, 0, n = 8*n + 1; (qfrep([1, 0; 0, 32], n) - qfrep([4, 2; 2, 9], n))[n])}; /* Michael Somos, Sep 02 2006 */
(Magma) Basis( CuspForms( Gamma1(128), 1), 641)[1]; /* Michael Somos, Jan 31 2015 */

Crossrefs

Cf. A000203, A002513, A107063, A107064, A138514.

Sequence in context: A251635 A156319 A190893 * A083650 A138514 A286137

Adjacent sequences: A030201 A030202 A030203 * A030205 A030206 A030207

Keyword

sign

Author

N. J. A. Sloane.

Status

approved