A161515 Expansion of q-series Sum_{n >= 0} (-1)^n*q^(n*(n + 1)/2)*(1 - q)*(1 - q^2)*...*(1 - q^n) / ((1 + q)*(1 + q^2)*(1 + q^3)*...*(1 + q^n)).

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

Offset

0,3

Comments

(-1)^n*a(n) is the number of inequivalent elements of norm 8n+1 in Z[sqrt(2)].

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Daniel Corson, David Favero, Kate Liesinger, Sarah Zubairy, Characters and q-series in Q(sqrt(2)), J. Number Theory, 107 (2004), 392-405.

Jeremy Lovejoy, Overpartitions and real quadratic fields, J. Number Theory, 106 (2004), 178-186.

Mathematica

CoefficientList[Series[Sum[(-1)^n*q^(Binomial[n+1, 2])*QPochhammer[q, q]^2*QPochhammer[q^(2n+2), q^2]/(QPochhammer[q^(n+1), q]^2* QPochhammer[q^2, q^2]), {n, 0, 50}], {q, 0, 80}], q] (* G. C. Greubel, Dec 04 2018 *)

Prog

(PARI) my(q='q+O('q^80)); Vec(sum(n=0, 100, (-1)^n*q^(binomial(n+1, 2))* prod(k=1, n, 1-q^k)/prod(j=1, n, 1+q^j))) \\ G. C. Greubel, Dec 04 2018
(Magma) m:=100; R<q>:=PowerSeriesRing(Integers(), m); [1] cat Coefficients(R!( (&+[(-1)^n*q^(Binomial(n+1, 2))*(&*[1-q^k: k in [1..n]])/(&*[1+q^j:j in [1..n]]): n in [1..100]]) )); // G. C. Greubel, Dec 04 2018
(Sage)
from sage.combinat.q_analogues import q_pochhammer
s=(sum( (-1)^n*x^(binomial(n+1, 2))*q_pochhammer(n, x, x)^2/q_pochhammer(n, x^2, x^2)
for n in range(80))).series(x, 80);
s.coefficients(x, sparse=False) # G. C. Greubel, Dec 04 2018

Crossrefs

Sequence in context: A189962 A308321 A097854 * A145580 A144219 A144027

Adjacent sequences: A161512 A161513 A161514 * A161516 A161517 A161518

Keyword

sign

Author

Jeremy Lovejoy, Jun 12 2009

Status

approved