A116498 Expansion of psi(-q)/psi(-q^2) in powers of q where psi() is a Ramanujan theta function.

1, -1, 1, -2, 1, -2, 3, -3, 4, -5, 6, -7, 8, -9, 11, -13, 16, -18, 21, -24, 27, -32, 36, -41, 48, -54, 61, -70, 78, -88, 100, -112, 127, -143, 159, -179, 199, -222, 248, -276, 308, -342, 380, -421, 465, -516, 570, -629, 697, -767, 845, -932, 1022, -1124, 1236, -1355, 1488, -1631, 1785, -1954, 2136

Offset: 0

Author

Michael Somos, Feb 18 2006

Keywords

• sign

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

Crossrefs

Cf. A000122, A000700, A010054, A070048, A121373.

Programs

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[ q^(1/8)* eta[q]*eta[q^4]^2/(eta[q^2]^2*eta[q^8]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 07 2018 *)

PARI

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^4+A)^2/eta(x^2+A)^2/eta(x^8+A), n))}

Formula

Expansion of q^(1/8)*eta(q)*eta(q^4)^2/(eta(q^2)^2*eta(q^8)) in powers of q.

a(n)=(-1)^n*A070048(n).

Euler transform of period 8 sequence [ -1,1,-1,-1,-1,1,-1,0,...].

Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x),B(x^3)) where f(u,v)=3*u*v -(u+v^3)*(v-u^3).

G.f.: Product_{k>0} (1+x^(2k))/((1+x^k)(1+x^(4k))) = (Sum_{k>0} (-x)^((k^2-k)/2))/(Sum_{k>0} (-x^2)^((k^2-k)/2)).