A113421 Expansion of eta(q)^2 * eta(q^4) * eta(q^6)^2 * eta(q^12) / eta(q^3)^2 in powers of q.

1, -2, -1, 4, -4, 2, 6, -8, 1, 8, -12, -4, 14, -12, 4, 16, -16, -2, 18, -16, -6, 24, -24, 8, 21, -28, -1, 24, -28, -8, 30, -32, 12, 32, -24, 4, 38, -36, -14, 32, -40, 12, 42, -48, -4, 48, -48, -16, 43, -42, 16, 56, -52, 2, 48, -48, -18, 56, -60, 16, 62, -60, 6, 64, -56, -24, 66, -64, 24, 48, -72, -8, 74, -76, -21, 72

Offset

1,2

Comments

Number 26 of the 74 eta-quotients listed in Table I of Martin (1996).

Links

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

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Formula

Euler transform of period 12 sequence [-2, -2, 0, -3, -2, -2, -2, -3, 0, -2, -2, -4, ...].

G.f.: Sum_{k>0} (3*k - 2) * x^(3*k - 2) / (1 + x^(6*k - 4)) - (3*k - 1) * x^(3*k - 1) / (1 + x^(6*k - 2)).

G.f.: Sum_{k>0} -(-1)^k * x^(2*k - 1) * (1 - x^(2*k - 1))^2 * (1 - x^(4*k - 2)) / (1 - x^(6*k - 3))^2.

a(n) is multiplicative with a(2^e) = (-2)^e, a(3^e) = (-1)^e, a(p^e) = (x^(e+1) - y^(e+1)) / (x - y) where x = p * Kronecker( -3, p) and y = (-1)^[p/2].

Example

G.f. = q - 2*q^2 - q^3 + 4*q^4 - 4*q^5 + 2*q^6 + 6*q^7 - 8*q^8 + q^9 + 8*q^10 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^2 QPochhammer[ q^4] QPochhammer[ q^6]^2 QPochhammer[ q^12] / QPochhammer[ q^3]^2, {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d * kronecker( -3, d) * (-1)^(n / d \ 2)))};
(PARI) {a(n) = my(A, p, e, t); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, (-2)^e, p==3, (-1)^e, t = (-1)^(p\2); p *= kronecker( -3, p); (p^(e+1) - t^(e+1)) / (p - t))))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^12 + A) / eta(x^3 + A)^2, n))};

Crossrefs

Sequence in context: A309086 A261070 A249140 * A135366 A247248 A192017

Adjacent sequences: A113418 A113419 A113420 * A113422 A113423 A113424

Keyword

sign,mult

Author

Michael Somos, Oct 29 2005

Status

approved