A131999 Expansion of eta(q)^2 * eta(q^2) * eta(q^4)^3 / eta(q^8)^2 in powers of q.

1, -2, -2, 4, -2, 8, 4, -16, -2, -14, 8, 20, 4, 24, -16, -16, -2, -36, -14, 36, 8, 32, 20, -48, 4, -42, 24, 40, -16, 56, -16, -64, -2, -40, -36, 64, -14, 72, 36, -48, 8, -84, 32, 84, 20, 56, -48, -96, 4, -114, -42, 72, 24, 104, 40, -80, -16, -72, 56, 116, -16

Offset

0,2

Comments

Number 19 of the 74 eta-quotients listed in Table I of Martin (1996).

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

References

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.67).

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of phi(q) * phi(q^2) * phi(-q)^2 in powers of q where phi() is a Ramanujan theta function.

Euler transform of period 8 sequence [-2, -3, -2, -6, -2, -3, -2, -4, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^4 + u^2*v^2 + 2 * u^2*w^2 + 2 * u*v*w * (-u + 2*v - 2*w) - 2 * u*v^3.

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

a(2*n) = a(n) for all n in Z.

G.f.: 1 - 2* Sum_{k>0} k * x^k / (1 - x^k) * Kronecker(2, k).

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

a(n) = -2 * A117000(n) unless n=0. a(n) = (-1)^n * A113416(n). a(2*n + 1) = - 2 * A113417(n).

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^(11/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A124340. - Michael Somos, Jun 28 2015

Convolution square is A259491. - Michael Somos, Jun 28 2015

Example

G.f. = 1 - 2*q - 2*q^2 + 4*q^3 - 2*q^4 + 8*q^5 + 4*q^6 - 16*q^7 - 2*q^8 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], -2 DivisorSum[ n, # KroneckerSymbol[ 2, #] &]]; (* Michael Somos, Jun 28 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^3 / QPochhammer[ q^8]^2, {q, 0, n}]; (* Michael Somos, Jun 28 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q]^2, {q, 0, n}]; (* Michael Somos, Jun 28 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^4]^2, {q, 0, n}]; (* Michael Somos, Jun 28 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, -2 * sumdiv(n, d, d * kronecker( 2, d)))};
(PARI) {a(n) = my(A, p, e); if( n<0, n==0, A = factor(n); -2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, abs(p%8-4)==3, (p^(e+1) - 1) / (p - 1), ((-p)^(e+1) - 1) / (-p - 1))))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A) * eta(x^4 + A)^3 / eta(x^8 + A)^2, n))};
(Magma) A := Basis( ModularForms( Gamma1(8), 2), 61); A[1] - 2*A[2] - 2*A[3] + 4*A[4] - 2*A[5]; /* Michael Somos, Jun 28 2015 */

Crossrefs

Cf. A113416, A113417, A117000, A124340, A259491.

Sequence in context: A259192 A307313 A338042 * A113416 A303140 A103178

Adjacent sequences: A131996 A131997 A131998 * A132000 A132001 A132002

Keyword

sign

Author

Michael Somos, Aug 06 2007

Status

approved