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A161516 Expansion of 1 + 2*Sum_{n >= 1} (-q)^n*(1 + q^2)*(1 + q^4)*...*(1 + q^(2*k - 2))/ ((1 - q)*(1 - q^3)*...*(1 - q^(2*n - 1))) 13
1, -2, 0, -2, 2, 0, 2, 0, 2, -2, 2, 0, 0, -2, 0, -4, 2, 0, 0, -2, 0, -2, 2, 0, 2, -2, 2, 0, 2, 0, 2, 0, 0, -2, 2, -2, 4, 0, 0, -2, 0, 0, 0, -4, 0, -2, 2, 0, 2, -4, 0, 0, 0, -2, 2, -2, 0, 0, 2, 0, 2, -2, 0, -2, 4, 0, 4, 0, 0, 0, 0, -2, 2, 0, 0, -2, 2, 0, 4, -2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(-1)^n*a(n) is the number of inequivalent elements of norm 8*n-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

With[{m=80}, CoefficientList[Series[1+2*Sum[(-q)^n*QPochhammer[q^4, q^4]*QPochhammer[q^(2*n+1), q]/((1+q^(2*n))*QPochhammer[q^(4*n+4), q^4]*QPochhammer[q, q]), {n, 1, m}], {q, 0, m}], q]] (* G. C. Greubel, Dec 04 2018 *)

PROG

(PARI) m=80; my(q='q+O('q^m)); Vec(1 + 2*sum(n=1, m, ((-q)^n/(1+q^(2*n) ))*prod(k=1, n, (1-q^(4*k))/((1-q^(2*k))*(1-q^(2*k-1)))) )) \\ G. C. Greubel, Dec 04 2018

(MAGMA) m:=80; R<q>:=PowerSeriesRing(Integers(), m); [1] cat Coefficients(R!( 2*(&+[((-q)^n/(1+q^(2*n)))*(&*[(1-q^(4*k))/((1-q^(2*k))*(1-q^(2*k-1))): k in [1..n]]): n in [1..m]]) )); // G. C. Greubel, Dec 04 2018

(Sage)

from sage.combinat.q_analogues import q_pochhammer

prec = 80

R = PowerSeriesRing(ZZ, 'x')

x = R.gen().O(prec)

s = 1+2*sum( (-x)^n*q_pochhammer(n, x^4, x^4)/((1+x^(2*n))* q_pochhammer(2*n, x, x)) for n in (1..prec))

print(s.coefficients()) # G. C. Greubel, Dec 04 2018

CROSSREFS

Sequence in context: A230205 A024713 A123530 * A329491 A123063 A031358

Adjacent sequences:  A161513 A161514 A161515 * A161517 A161518 A161519

KEYWORD

sign

AUTHOR

Jeremy Lovejoy, Jun 12 2009

STATUS

approved

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Last modified April 6 12:08 EDT 2020. Contains 333273 sequences. (Running on oeis4.)