A080966 Expansion of theta_4(q^2) * theta_2(q)^2/(4*q^(1/2)) in powers of q.

1, 2, -1, -2, 0, -4, -1, 2, -4, 2, 4, 2, 1, -2, 4, 2, 4, 0, -4, 0, -3, 4, -4, -4, 0, -2, 0, -6, 0, 2, -1, -4, 4, -4, -4, 8, 4, 6, 0, 2, -8, 0, 7, 2, 4, 2, 4, 0, 0, -6, 4, 0, -4, 0, 0, 0, 1, -6, -4, 4, -8, -2, -4, 4, 0, 2, -4, -6, 0, -2, 4, -8, 1, 2, 0, 0, 4, 4, 4, -2, 4, 6, 0, -2, 0, -4, -8, 10, 8, 8, -1, 4, 4, 2, -4, -4, -8, 6, 4, -6, 8, -6, 4, 4

Offset

0,2

Comments

The nonzero quadrisection of A248395.

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..5000

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.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

G.f.: Product_{k>0} (1+x^k)^2*(1-x^(2k))^3/(1+x^(2k)).

Expansion of f(-q^4)*f(q)^2 in powers of q where f(-q)=f(-q,-q^2) is a Ramanujan theta function.

Expansion of q^(-1/4)*eta(q^2)^6/(eta(q)^2*eta(q^4)) in powers of q.

Euler transform of period-4 sequence [2,-4,2,-3,...].

G.f.: Product_{k>0} (1-x^(2*k))^3*(1+x^k)^2/(1+x^(2*k)).

2*a(n) = A080964(4*n+1) = 2*A072071(4*n+1) - A072070(4*n+1).

Example

q + 2*q^5 - q^9 - 2*q^13 - 4*q^21 - q^25 + 2*q^29 - 4*q^33 + ...

Mathematica

QP = QPochhammer; s = QP[q^2]^6/(QP[q]^2*QP[q^4]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
QP := QPochhammer; a:=CoefficientList[Series[QP[q^2]^6/(QP[q]^2*QP[q^4]), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 01 2018 *)

Prog

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^6/eta(x+A)^2/eta(x^4+A), n))}

Crossrefs

Sequence in context: A132456 A257873 A229817 * A187150 A023895 A070963

Adjacent sequences: A080963 A080964 A080965 * A080967 A080968 A080969

Keyword

sign

Author

Michael Somos, Feb 28 2003

Status

approved