A122864 Expansion of eta(q^3)^2 * eta(q^4) * eta(q^6)^2 * eta(q^36) / (eta(q) * eta(q^9) * eta(q^12)^2) in powers of q.

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

Offset

1,3

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

Euler transform of period 36 sequence [ 1, 1, -1, 0, 1, -3, 1, 0, 0, 1, 1, -2, 1, 1, -1, 0, 1, -2, 1, 0, -1, 1, 1, -2, 1, 1, 0, 0, 1, -3, 1, 0, -1, 1, 1, -2, ...].

Moebius transform is period 36 sequence [ 1, 0, 1, 0, 1, 0, -1, 0, -4, 0, -1, 0, 1, 0, 1, 0, 1, 0, -1, 0, -1, 0, -1, 0, 1, 0, 4, 0, 1, 0, -1, 0, -1, 0, -1, 0, ...].

a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2*(-1)^(e+1) if e>0, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).

a(3*n) = 2 * A113446(n). a(3*n + 1) = A002564(3*n + 1) = A035154(3*n + 1) = A113446(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Oct 15 2022

Example

q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 + q^8 - 2*q^9 + 2*q^10 + 2*q^12 + ...

Mathematica

eta[x_] := x^(1/24)*QPochhammer[x]; A122864[n_] := SeriesCoefficient[ (eta[q^3]^2*eta[q^4]*eta[q^6]^2*eta[q^36])/(eta[q]*eta[q^9]*eta[q^12]^2), {q, 0, n}]; Table[A122864[n], {n, 50}] (* G. C. Greubel, Sep 16 2017 *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^36 + A) /(eta(x + A) * eta(x^9 + A) * eta(x^12 + A)^2), n))}
(PARI) {a(n) = local(A, p, e); if( n<1, 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, -2*(-1)^e, if( p%4==1, e+1, 1-e%2))))))}

Crossrefs

Cf. A002564, A019670, A035154, A113446, A122856, A122865.

Sequence in context: A073267 A159981 A071858 * A140084 A243747 A105937

Adjacent sequences: A122861 A122862 A122863 * A122865 A122866 A122867

Keyword

sign,mult

Author

Michael Somos, Sep 15 2006

Status

approved