A143160 Expansion of q^(-5/12) * eta(q) * eta(q^2) * eta(q^3) * eta(q^4) in powers of q.

1, -1, -2, 0, 0, 5, 1, 1, -2, -7, 4, -5, -2, -1, 4, 7, -1, 5, -1, 2, 2, 4, -13, -10, 1, -1, -2, 3, 6, -8, -1, 2, 9, 4, 9, 3, -1, -3, 9, -8, -9, 2, -9, 3, -12, -10, 1, 11, -6, 14, -11, -1, 1, 2, 18, -13, 3, 12, 13, 6, 6, -7, -3, -5, -2, -14, 2, -10, -7, -2, -18

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Formula

Euler transform of period 12 sequence [-1, -2, -2, -3, -1, -3, -1, -3, -2, -2, -1, -4, ...].

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

Example

G.f. = 1 - x - 2*x^2 + 5*x^5 + x^6 + x^7 - 2*x^8 - 7*x^9 + 4*x^10 - 5*x^11 + ...
G.f. = q^5 - q^17 - 2*q^29 + 5*q^65 + q^77 + q^89 - 2*q^101 - 7*q^113 + ...

Maple

N:= 100: # for a(0)..a(N)
g:= mul(1-x^k, k=1..N)*mul(1-x^(2*k), k=1..N/2)*mul(1-x^(3*k), k=1..N/3)*mul(1-x^(4*k), k=1..N/4):
S:= series(g, x, N+1):
seq(coeff(S, x, n), n=0..N); # Robert Israel, Feb 08 2018

Mathematica

a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-1/8) QPochhammer[ x^2]^2 EllipticTheta[ 2, Pi/4, x^(1/2)] QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ 2^(-1) x^(-1/4) QPochhammer[ -x] EllipticTheta[ 2, Pi/4, x^(1/2)]^2 QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^4 + A), n))};

Crossrefs

Sequence in context: A275744 A290321 A145430 * A369730 A156387 A370065

Adjacent sequences: A143157 A143158 A143159 * A143161 A143162 A143163

Keyword

sign

Author

Michael Somos, Jul 27 2008

Status

approved