A128144 Expansion of chi(-q)* chi(-q^2)* chi(-q^9)/( chi(-q^3)* chi(q^9)) in powers of q where chi() is a Ramanujan theta function.

1, -1, -1, 1, 0, -1, 0, 1, 1, -2, 0, 3, 0, -2, 0, 3, 0, -5, 0, 4, -2, -4, 0, 5, 0, -7, 2, 7, 0, -5, 0, 10, 1, -12, 0, 10, 0, -14, -4, 17, 0, -21, 0, 22, 4, -24, 0, 34, 0, -33, 1, 36, 0, -45, 0, 45, -8, -52, 0, 55, 0, -62, 8, 71, 0, -70, 0, 88, 2, -96, 0, 98, 0, -122, -14, 133, 0, -148, 0, 163, 14, -182, 0, 217, 0, -216

Offset

0,10

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

Expansion of (eta(q)* eta(q^6)* eta(q^36)* eta(q^9)^2)/(eta(q^3)* eta(q^4)* eta(q^18)^3) in powers of q.

Euler transform of period 36 sequence [ -1, -1, 0, 0, -1, -1, -1, 0, -2, -1, -1, 0, -1, -1, 0, 0, -1, 0, -1, 0, 0, -1, -1, 0, -1, -1, -2, 0, -1, -1, -1, 0, 0, -1, -1, 0, ...].

G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)= (1-v)*(1-v+v^2)*(2*u-u^2)^2 -(u+v-u*v)^2*(u-v)^2.

a(6*n+4)=0. a(6*n)=0 if n>0.

A092848(n) = -a(6*n+2).

A128143(n) = -a(n) if n>0.

A128145(n) = -a(n) if n>0.

Mathematica

A128144[n_] := SeriesCoefficient[((QPochhammer[q]*QPochhammer[q^6] *QPochhammer[q^36]*QPochhammer[q^9]^2)/(QPochhammer[q^3]*QPochhammer[q^4] *QPochhammer[q^18]^3)), {q, 0, n}]; Table[A128144[n], {n, 0, 50}] (* G. C. Greubel, Oct 09 2017 *)

Prog

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^6+A)*eta(x^36+A)*eta(x^9+A)^2/ (eta(x^3+A)*eta(x^4+A)*eta(x^18+A)^3), n))}

Crossrefs

Sequence in context: A256580 A213266 A182038 * A128145 A128143 A292561

Adjacent sequences: A128141 A128142 A128143 * A128145 A128146 A128147

Keyword

sign

Author

Michael Somos, Feb 16 2007

Status

approved