A122861 - OEIS

A122861

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

1, -3, 2, 0, 2, -3, 2, 0, 1, -6, 2, 0, 2, 0, 2, 0, 3, -6, 0, 0, 2, -3, 2, 0, 2, -6, 2, 0, 0, 0, 4, 0, 2, -3, 2, 0, 2, -6, 0, 0, 1, -6, 2, 0, 4, 0, 2, 0, 0, -6, 2, 0, 2, 0, 2, 0, 3, -6, 2, 0, 2, 0, 0, 0, 2, -9, 2, 0, 0, -6, 2, 0, 4, 0, 2, 0, 2, 0, 0, 0, 2, -6, 4, 0, 0, -3, 4, 0, 0, -6, 2, 0, 2, 0, 2, 0, 1, -6, 0, 0, 4, -6, 2, 0, 2

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OFFSET

0,2

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, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

FORMULA

Expansion of q^(-1/3)*eta(q)^3*eta(q^6)^2/(eta(q^2)^2*eta(q^3)) in powers of q.

Euler transform of period 6 sequence [ -3, -1, -2, -1, -3, -2, ...].

a(n) = b(3n+1) where b(n) is multiplicative and b(2^e) = -3(1+(-1)^e)/2 if e>0, b(3^e) = 0^e, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).

a(4n+3) = 0.

a(n) = A115979(3n+1) = A097109(3n+1).

a(2n) = A097195(n) = A033687(2n); a(2n+1) = -3*A033687(2n+1).

a(n) = (-1)^n * A129576(n). - Amiram Eldar, Jan 28 2024

MATHEMATICA

A122861[n_] := SeriesCoefficient[(QPochhammer[q]^3*QPochhammer[q^6]^2)/(QPochhammer[q^2]^2 *QPochhammer[q^3]), {q, 0, n}]; Table[A122861[n], {n, 0, 50}] (* G. C. Greubel, Oct 05 2017 *)

PROG

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

(PARI) {a(n)=local(A, p, e); if(n<0, 0, n=3*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 3*(e%2-1), if(p==3, 0, if(p%6==1, e+1, !(e%2)))))))}

CROSSREFS

Cf. A033687, A097109, A097195, A115979, A129576.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A227864 A236603 A129576 * A326045 A277097 A077814

Adjacent sequences: A122858 A122859 A122860 * A122862 A122863 A122864

KEYWORD

sign

AUTHOR

Michael Somos, Sep 15 2006

STATUS

approved