A082564 Expansion of eta(q)^2 * eta(q^2) / eta(q^4) in powers of q.

1, -2, -2, 4, 2, 0, -4, 0, 2, -6, 0, 4, 4, 0, 0, 0, 2, -4, -6, 4, 0, 0, -4, 0, 4, -2, 0, 8, 0, 0, 0, 0, 2, -8, -4, 0, 6, 0, -4, 0, 0, -4, 0, 4, 4, 0, 0, 0, 4, -2, -2, 8, 0, 0, -8, 0, 0, -8, 0, 4, 0, 0, 0, 0, 2, 0, -8, 4, 4, 0, 0, 0, 6, -4, 0, 4, 4, 0, 0, 0, 0, -10, -4, 4, 0, 0, -4, 0, 4, -4, 0, 0, 0, 0, 0, 0, 4, -4, -2, 12, 2, 0, -8, 0

Offset

0,2

Comments

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

a(n) is nonzero if and only if n is in A002479. - Michael Somos, Dec 15 2011

Absolute values appear to give A033715 = 2*A002325.

Denoted by a_4(n) in Kassel and Reutenauer 2015. - Michael Somos, Jun 04 2015

Links

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

Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015 , see page 31 7.2(c). [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]

Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016, see p. 13 paragraph 3.3.3.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of phi(-q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 30 2007

Euler transform of period 4 sequence [ -2, -3, -2, -2, ...]. - Michael Somos, Mar 30 2007

G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(11/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033761. - Michael Somos, Aug 29 2014

G.f.: Product_{k>0} (1 - x^k)^2 / (1 + x^(2*k)). - Michael Somos, Mar 30 2007

a(n) = -2 * A129134(n) unless n=0. - Michael Somos, Mar 30 2007

a(n) = (-1)^floor( (n+1)/2 ) * A033715(n). - Michael Somos, Aug 29 2014

a(2*n) = A133692(n). a(2*n + 1) = -2 * A125095(n). - Michael Somos, Aug 29 2014

a(3*n + 1) = -2 * A258747(n). a(3*n + 2) = -2 * A258764(n). - Michael Somos, Jun 09 2015

Example

G.f. = 1 - 2*q - 2*q^2 + 4*q^3 + 2*q^4 - 4*q^6 + 2*q^8 - 6*q^9 + 4*q^11 + 4*q^12 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^2] / QPochhammer[ q^4], {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2], {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 2 (-1)^Quotient[ n + 1, 2] DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* Michael Somos, Aug 29 2014 *)

Prog

(PARI) {a(n) = if( n<1, n==0, 2 * (-1)^((n+1) \ 2) * sumdiv( n, d, kronecker( -2, d)))}; /* Michael Somos, Mar 30 2007 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A) / eta(x^4 + A), n))};
(Magma) A := Basis( ModularForms( Gamma1(32), 1), 105); A[1] - 2*A[2] - 2*A[3] + 4*A[4] + 2*A[5] - 4*A[7] + 2*A[9] - 6*A[10] + 4*A[12] + 4*A[13] - 4*A[16]; /* Michael Somos, Aug 29 2014 */

Crossrefs

Cf. A002479, A033715, A033761, A125095, A129134, A133692, A258747, A258764.

Sequence in context: A080963 A133692 A033715 * A139093 A080918 A033758

Adjacent sequences: A082561 A082562 A082563 * A082565 A082566 A082567

Keyword

sign

Author

Benoit Cloitre, May 05 2003

Status

approved