A096920 Expansion of q^(-1/12) * eta(q^2)^4 / (eta(q)^2 * eta(q^4)) in powers of q.

1, 2, 1, 2, 3, 2, 4, 4, 4, 6, 7, 8, 8, 10, 11, 14, 16, 16, 20, 22, 24, 28, 32, 34, 39, 44, 48, 54, 60, 66, 73, 82, 88, 98, 108, 118, 132, 144, 156, 172, 188, 204, 224, 244, 265, 290, 316, 340, 372, 404, 436, 474, 513, 554, 600, 650, 700, 756, 816, 878, 948, 1022, 1096, 1182

Offset

0,2

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Links

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

a(n) = b(n)+b(n-1)+b(n-3)+b(n-6)+...+b(n-k*(k+1)/2)+..., where b() is A000700(). E.g., a(8) = b(8)+b(7)+b(5)+b(2) = 2+1+1+0 = 4.

G.f.: Product_{k>0} (1 - x^(4*k)) * (1 + x^(2*k-1))^2. - Michael Somos, Mar 25 2008

Expansion of psi(q) * chi(q) = f(q) / chi(-q) = f(q)^2 / phi(-q^2) = phi(-q^2) / chi(-q)^2 = phi(q) / chi(-q^2) = psi(q)^2 / f(-q^4) = f(-q^4) * chi(q)^2 where phi(), psi(), chi(), f() are Ramanujan theta functions.

Euler transform of period 4 sequence [ 2, -2, 2, -1, ...]. - Michael Somos, Mar 25 2008

G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 12^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138559.

a(n) ~ exp(Pi*sqrt(n/6)) / (2*sqrt(2*n)). - Vaclav Kotesovec, Sep 07 2015

Example

q + 2*q^13 + q^25 + 2*q^37 + 3*q^49 + 2*q^61 + 4*q^73 + 4*q^85 + 4*q^97 + ...

Mathematica

nmax = 40; CoefficientList[Series[Product[(1 - x^(4*k)) * (1 + x^(2*k-1))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/12) *eta[q^2]^4/(eta[q]^2*eta[q^4]), {q, 0, 50}], q]; Table[a[[n]], {n, 0, 50}] (* G. C. Greubel, May 09 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 / eta(x + A)^2 / eta(x^4 + A), n))} /* Michael Somos, Mar 25 2008 */

Crossrefs

Sequence in context: A080845 A290370 A029166 * A087154 A029839 A082304

Adjacent sequences: A096917 A096918 A096919 * A096921 A096922 A096923

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 18 2004

Status

approved