A113419 Expansion of phi(x)^2 * phi(-x) * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

1, 2, -4, -8, 7, 10, -12, -8, 18, 18, -16, -24, 21, 20, -28, -32, 20, 32, -36, -24, 42, 42, -28, -48, 57, 36, -52, -40, 36, 58, -60, -56, 48, 66, -48, -72, 74, 42, -80, -80, 61, 82, -72, -56, 90, 96, -64, -72, 98, 70, -100, -104, 64, 106, -108, -72, 114, 96

Offset

0,2

Comments

Number 46 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).

Links

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/2) * (eta(q^2)^9 * eta(q^8)^2) / (eta(q)^2 * eta(q^4)^5) in powers of q. - Michael Somos, Mar 14 2012

Euler transform of period 8 sequence [ 2, -7, 2, -2, 2, -7, 2, -4, ...].

a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (x^(e+1) - y^(e+1)) / (x - y) where x = p * (-1)^floor(p/4) and y = (-1)^floor(p/2).

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

a(n) = (-1)^n * A113417(n) = (-1)^floor(n/2) * A258096(n) = (-1)^(n + floor(n/2)) * A209940(n).

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A109039. - Michael Somos, May 20 2015

Example

G.f. = 1 + 2*x - 4*x^2 - 8*x^3 + 7*x^4 + 10*x^5 - 12*x^6 - 8*x^7 + 18*x^8 + ...
G.f. = q + 2*q^3 - 4*q^5 - 8*q^7 + 7*q^9 + 10*q^11 - 12*q^13 - 8*q^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^9 QPochhammer[ q^8]^2 / (QPochhammer[ q]^2 QPochhammer[ q^4]^5), {q, 0, n}]; (* Michael Somos, May 19 2015 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 2 n + 1, KroneckerSymbol[ 2, #] # &]]; (* Michael Somos, May 19 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, d*(d%2) * (-1)^((n/d)\2 + (d-1)\4)))};
(PARI) {a(n) = my(A, p, e, t); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, t = (-1)^(p\2); p *= kronecker( -2, p); (p^(e+1) - t^(e+1)) / (p - t) )))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^5), n))};

Crossrefs

Cf. A109039, A113417, A209940, A258096.

Sequence in context: A343629 A039794 A113417 * A209940 A258096 A261298

Adjacent sequences: A113416 A113417 A113418 * A113420 A113421 A113422

Keyword

sign

Author

Michael Somos, Oct 29 2005

Status

approved