A112301 - OEIS

%I #23 Mar 12 2021 22:24:43

%S 1,-2,0,0,2,0,0,0,1,-4,0,0,2,0,0,0,2,-2,0,0,0,0,0,0,3,-4,0,0,2,0,0,0,

%T 0,-4,0,0,2,0,0,0,2,0,0,0,2,0,0,0,1,-6,0,0,2,0,0,0,0,-4,0,0,2,0,0,0,4,

%U 0,0,0,0,0,0,0,2,-4,0,0,0,0,0,0,1,-4,0,0,4,0,0,0,2,-4,0,0,0,0,0,0,2,-2,0,0,2,0,0,0,0

%N Expansion of (eta(q) * eta(q^16))^2 / (eta(q^2) * eta(q^8)) in powers of q.

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

%H G. C. Greubel, <a href="/A112301/b112301.txt">Table of n, a(n) for n = 1..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q * phi(-q) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.

%F Expansion of (phi(-q^2)^2 - phi(-q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

%F Euler transform of period 16 sequence [ -2, -1, -2, -1, -2, -1, -2, 0, -2, -1, -2, -1, -2, -1, -2, -2, ...].

%F a(n) is ultiplicative with a(2) = -2, a(2^e) = 0 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4).

%F Moebius transform is period 16 sequence [ 1, -3, -1, 2, 1, 3, -1, 0, 1, -3, -1, -2, 1, 3, -1, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 4 (t/i) f(t) where q = exp(2 Pi i t).

%F G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 + x^(8*k))^2 * (1 + x^(2*k)) * (1 + x^(4*k)).

%F G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)).

%F a(4*n) = a(4*n + 3) = a(8*n + 6) = 0. a(8*n + 2) = -2 * A008441(n).

%F a(n) = -(-1)^n * A134013(n). a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n).

%e G.f. = q - 2*q^2 + 2*q^5 + q^9 - 4*q^10 + 2*q^13 + 2*q^17 - 2*q^18 + 3*q^25 - ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q^4] / 2, {q, 0, n}]; (* _Michael Somos_, Oct 19 2013 *)

%t QP = QPochhammer; s = (QP[q]*QP[q^16])^2/(QP[q^2]*QP[q^8]) + O[q]^100; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015 *)

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^16 + A))^2 / (eta(x^2 + A) * eta(x^8 + A)), n))};

%o (PARI) {a(n) = if( n>0 & (n+1)%4\2, (n%2*3 - 2) * sumdiv( n / gcd(n, 2), d, (-1)^(d\2)))};

%Y Cf. A008441, A053692, A113407, A134013.

%K sign,mult

%O 1,2

%A _Michael Somos_, Sep 02 2005, Oct 02 2007