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A133690 Expansion of (phi(-q) * phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function. 4
1, -4, 8, -16, 24, -24, 32, -32, 24, -52, 48, -48, 96, -56, 64, -96, 24, -72, 104, -80, 144, -128, 96, -96, 96, -124, 112, -160, 192, -120, 192, -128, 24, -192, 144, -192, 312, -152, 160, -224, 144, -168, 256, -176, 288, -312, 192, -192, 96, -228, 248, -288 (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).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..5000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2))^2 in powers of q.

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

a(n) = -4 * b(n) where b() is multiplicative with b(2) = -2, b(2^e) = -6 if e>1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133657.

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

a(n) = (-1)^n * A097057(n). Convolution square of A133692.

a(2*n) = 8 * A046897(n) unless n=0. a(2*n + 1) = A008438(n). a(4*n) = A004011(n). a(4*n + 1) = -4 * A112610(n). a(4*n + 3) = -16 * A097723(n).

EXAMPLE

G.f. = 1 - 4*q + 8*q^2 - 16*q^3 + 24*q^4 - 24*q^5 + 32*q^6 - 32*q^7 + 24*q^8 - ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)

a[ n_] := If[ n < 1, Boole[n == 0], -4 Which[ OddQ[n], DivisorSigma[ 1, n], Mod[n, 4] > 0, -2 DivisorSigma[1, n/2], True, -6 DivisorSum[n/4, # Mod[#, 2] &]]]; (* Michael Somos, Oct 30 2015 *)

PROG

(PARI) {a(n) = if( n<1, n==0, -4 * if( n%2, sigma(n), n%4, -2 * sigma(n/2), -6 * sumdiv( n/4, d, (d%2)*d )))};

(PARI) {a(n) = local(A); if ( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A)^5 / (eta(x^2 + A)^3 * eta(x^8 + A)^2))^2, n))};

CROSSREFS

Cf. A004011, A008438, A097057, A097723, A112610, A133657, A133692.

Sequence in context: A036302 A032377 A312823 * A097057 A333168 A306219

Adjacent sequences:  A133687 A133688 A133689 * A133691 A133692 A133693

KEYWORD

sign

AUTHOR

Michael Somos, Sep 20 2007

STATUS

approved

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Last modified November 27 06:49 EST 2020. Contains 338678 sequences. (Running on oeis4.)