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A226139 Expansion of b(-q) * b(q^2) in powers of q where b() is a cubic AGM theta function. 2
1, 3, -3, -15, -3, 18, 15, 24, -3, -69, -18, 36, 15, 42, -24, -90, -3, 54, 69, 60, -18, -120, -36, 72, 15, 93, -42, -231, -24, 90, 90, 96, -3, -180, -54, 144, 69, 114, -60, -210, -18, 126, 120, 132, -36, -414, -72, 144, 15, 171, -93, -270, -42, 162, 231, 216 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 21 of the 126 eta-quotients listed in Table 1 of Williams 2012.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.
FORMULA
Expansion of = b(q^2) * (2 * b(q^4) - b(q)) = b(q^2)^4 / (b(q) * b(q^4)) in powers of q where b() is a cubic AGM function.
Expansion of (a(q) + a(q^2)) * (a(q^2) - 2 * a(q^4)) / 2 in powers of q where a() is a cubic AGM theta function.
Expansion of (psi(q)^3 / psi(q^3)) * (phi(-q^2)^3 / phi(-q^6)) in power of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2)^12 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)^4) in powers of q.
a(n) = 3 * b(n) where b(n) is multiplicative and b(2^e) = -1, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226132.
Euler transform of period 12 sequence [ 3, -9, 2, -6, 3, -6, 3, -6, 2, -9, 3, -4, ...].
G.f.: 1 + 3 * (Sum_{k>0} x^k / (1 - (-x)^k)^2 - 9 * x^(3*k) / (1 - (-x)^(3*k))^2).
G.f.: 1 + 3 * Sum_{k>0 not 2|k} k * (x^k * x^k / (1 + x^(2*k)) - 9 * x^(3*k) / (1 + x^(3*k))).
G.f.: Product_{k>0} (1 - x^(2*k))^12 * (1 + x^(6*k)) / ( (1 - x^k)^3 * (1 + x^(3*k))^3 * (1 - x^(3*k))^2 * (1 - x^(4*k))^3 ).
a(n) = (-1)^n * A131943(n). a(2*n) = A131943(n). a(2*n + 1) = 3 * A134077(n).
EXAMPLE
1 + 3*q - 3*q^2 - 15*q^3 - 3*q^4 + 18*q^5 + 15*q^6 + 24*q^7 - 3*q^8 + ...
MATHEMATICA
a[ n_] := If[ n < 1, Boole[n == 0], - 3 (-1)^n Sum[ d {0, 1, 0, -2, 0, 1}[[Mod[ d, 6] + 1]], {d, Divisors @n}]]
a[ n_] := If[ n < 2, Boole[n == 0] + 3 Boole[n == 1], 3 Times @@ (Which[ # == 2, -1, # == 3, 4 - 3 #^#2, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[n])]
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^2]^3 / EllipticTheta[ 4, 0, q^6] EllipticTheta[ 2, 0, q^(1/2)]^3 / EllipticTheta[ 2, 0, q^(3/2)] / 4, {q, 0, n}]
PROG
(PARI) {a(n) = if( n<1, n==0, -3 * (-1)^n * sumdiv( n, d, d * [0, 1, 0, -2, 0, 1][d%6 + 1]))}
(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 3 * prod( k= 1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, -1, if( p==3, 4 - p^(e+1), (p^(e+1) - 1) / (p - 1))))))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^12 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^4), n))}
CROSSREFS
Sequence in context: A281213 A282388 A131943 * A260119 A368117 A282009
KEYWORD
sign
AUTHOR
Michael Somos, May 27 2013
STATUS
approved

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Last modified March 28 11:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)