A124452 Expansion of psi(-q) * psi(-q^2) * chi(q^3) * chi(q^6) in powers of q where psi(), chi() are Ramanujan theta functions.

1, -1, -1, 1, -1, 0, 1, 0, -1, 3, 0, -2, 1, 0, 0, 0, -1, -2, 3, -2, 0, 0, -2, 0, 1, -1, 0, 5, 0, 0, 0, 0, -1, 2, -2, 0, 3, 0, -2, 0, 0, -2, 0, -2, -2, 0, 0, 0, 1, -1, -1, 2, 0, 0, 5, 0, 0, 2, 0, -2, 0, 0, 0, 0, -1, 0, 2, -2, -2, 0, 0, 0, 3, -2, 0, 1, -2, 0, 0, 0, 0, 7, -2, -2, 0, 0, -2, 0, -2, -2, 0, 0, 0, 0, 0, 0, 1, -2, -1, 6, -1, 0, 2, 0, 0

Offset

0,10

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

a(n) is nonzero if and only if n is in A002479.

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) * eta(q^6) * eta(q^8) * eta(q^12) / (eta(q^3) * eta(q^24)) in powers of q.

a(n) = -b(n) where b(n) is multiplicative with b(2^e) = 1, b(3^e) = 1 - 2*e, b(p^e) = 1+e if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8).

Euler transform of period 24 sequence [ -1, -1, 0, -1, -1, -1, -1, -2, 0, -1, -1, -2, -1, -1, 0, -2, -1, -1, -1, -1, 0, -1, -1, -2, ...].

Moebius transform is period 24 sequence [ -1, 0, 2, 0, 1, 0, 1, 0, 2, 0, -1, 0, 1, 0, -2, 0, -1, 0, -1, 0, -2, 0, 1, 0, ...].

a(8*n + 5) = a(8*n + 7) = 0. a(2*n) = a(n). a(3*n) >= 0.

Example

1 - q - q^2 + q^3 - q^4 + q^6 - q^8 + 3*q^9 - 2*q^11 + q^12 - q^16 - 2*q^17 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]* eta[q^6]*eta[q^8]*eta[q^12]/(eta[q^3]*eta[q^24]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 08 2018 *)

Prog

(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); - prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 1, if( p==3, 1 - 2*e, if( p%8<4, e+1, !(e%2)))))))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A) * eta(x^8 + A) * eta(x^12 + A) / (eta(x^3 + A) * eta(x^24 + A)), n))}
(PARI) q='q+O('q^99); Vec(eta(q)*eta(q^6)*eta(q^8)*eta(q^12)/(eta(q^3)*eta(q^24))) \\ Altug Alkan, Mar 09 2018

Crossrefs

Cf. A002479.

Sequence in context: A073538 A022898 A072780 * A351532 A291786 A004603

Adjacent sequences: A124449 A124450 A124451 * A124453 A124454 A124455

Keyword

sign

Author

Michael Somos, Nov 02 2006

Status

approved