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A121456 Expansion of q*(psi(-q)psi(-q^3))^2 in powers of q where psi() is a Ramanujan theta function. 1
1, -2, 1, -4, 6, -2, 8, -8, 1, -12, 12, -4, 14, -16, 6, -16, 18, -2, 20, -24, 8, -24, 24, -8, 31, -28, 1, -32, 30, -12, 32, -32, 12, -36, 48, -4, 38, -40, 14, -48, 42, -16, 44, -48, 6, -48, 48, -16, 57, -62, 18, -56, 54, -2, 72, -64, 20, -60, 60, -24, 62, -64, 8, -64, 84, -24, 68, -72, 24, -96, 72, -8, 74, -76, 31, -80 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

LINKS

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

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

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

Multiplicative with a(2^e) = -(2^e) if e>0, a(3^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p>3.

a(3n)=a(n), a(4n+2)=-2a(2n+1).

a(n) = (-1)^(n+1)*A111932(n).

MATHEMATICA

eta[q_]:= q^(1/24)*QPochhammer[q];  a[n_]:= SeriesCoefficient[(eta[q] *eta[q^3]*eta[q^4]*eta[q^12])^2/(eta[q^2]*eta[q^6])^2, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Mar 07 2018 *)

PROG

(PARI) {a(n)=if(n<1, 0, -(-1)^n*sumdiv(n, d, (n/d%2)*d*(d%3>0)))}

CROSSREFS

Sequence in context: A187141 A165604 A111932 * A193818 A127535 A285491

Adjacent sequences:  A121453 A121454 A121455 * A121457 A121458 A121459

KEYWORD

sign,mult

AUTHOR

Michael Somos, Jul 30 2006

STATUS

approved

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Last modified November 16 12:40 EST 2018. Contains 317272 sequences. (Running on oeis4.)