A123759 Expansion of f(-q)*psi(-q^5) in powers of q where f(), psi() are Ramanujan theta functions.

1, -1, -1, 0, 0, 0, 1, 2, 0, 0, -1, 0, -2, 0, 0, -2, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, -2, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, -2, 0, 0, -2, -1, 0, 0, 0, -2, 0, 0, 0, 0, 0, 2, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0

Offset

0,8

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} 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

Euler transform of period 20 sequence [ -1, -1, -1, -1, -2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, -1, -1, -1, -1, -2, ...].

Product_{k>0} (1-x^k)*(1-x^(5k))*(1+x^(10k)).

a(8n+3) = a(8n+5) = 0.

Expansion of q^(-2/3) * eta(q) * eta(q^5) * eta(q^20)/ eta(q^10) in powers of q.

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-2/3)* eta[q]*eta[q^5]*eta[q^20]/eta[q^10], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 08 2018 *)

Prog

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^5+A)*eta(x^20+A)/eta(x^10+A), n))}
(PARI) {a(n) = local(s, k); if(n<0, 0, n=24*n+16; forstep(k=1, sqrtint(n\15), 2, if(issquare(n-15*k^2, &j)& (j^2%6==1), s+= (-1)^((j+1)\6+ (k+2)\4))); s)}

Crossrefs

Sequence in context: A091400 A129448 A239003 * A072453 A307303 A324252

Adjacent sequences: A123756 A123757 A123758 * A123760 A123761 A123762

Keyword

sign

Author

Michael Somos, Oct 12 2006

Status

approved