A128145 Expansion of psi(q^3)* phi(-q^3)* chi^2(-q^3)/( psi(-q)* phi(-q^18)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

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) := Product_{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) := Product_{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^2)* eta(q^3)^3* eta(q^36))/(eta(q)* eta(q^4)* eta(q^6)* eta(q^18)^2) in powers of q.

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

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v - 1)*(3 - 3*v + v^2)*(2*u - u^2)^2 - (u + v - u*v)^2*(u - v)^2.

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

Mathematica

eta[x_] := x^(1/24)*QPochhammer[x]; A128145[n_] := SeriesCoefficient[ eta[q^2]*eta[q^3]^3*eta[q^36]/(eta[q]*eta[q^4]*eta[q^6]*eta[q^18]^2 ), {q, 0, n}]; Table[A128145[n], {n, 0, 50}] (* G. C. Greubel, Aug 16 2017 *)

Prog

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

Crossrefs

A092848(n) = a(6n+2). A128143(n) = a(n) if n > 0. A128144(n) = -a(n) if n > 0.

Sequence in context: A213266 A182038 A128144 * A128143 A292561 A027640

Adjacent sequences: A128142 A128143 A128144 * A128146 A128147 A128148

Keyword

sign

Author

Michael Somos, Feb 16 2007

Status

approved