A128143 Expansion of q* (psi(q^9)/phi(q^9))/ (psi(q)/phi(q)) in powers of q where psi(),phi() are Ramanujan theta functions.

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, -4

Offset

1,9

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 = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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

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

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

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

A128144(n) = -a(n) if n>0.

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

Mathematica

A128143[n_] := SeriesCoefficient[q*(QPochhammer[q^9]/QPochhammer[q])* (QPochhammer[q^36]/QPochhammer[q^4])^2*(QPochhammer[q^2]/QPochhammer[q^18])^3, {q, 0, n}]; Rest[Table[A128143[n], {n, 0, 1000}]] (* G. C. Greubel, Oct 09 2017 *)

Prog

(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x^2+A)^3*eta(x^9+A)*eta(x^36+A)^2/ (eta(x+A)*eta(x^4+A)^2*eta(x^18+A)^3), n))}

Crossrefs

Sequence in context: A182038 A128144 A128145 * A292561 A027640 A349448

Adjacent sequences: A128140 A128141 A128142 * A128144 A128145 A128146

Keyword

sign,changed

Author

Michael Somos, Feb 16 2007

Status

approved