A208664 - OEIS

%I #30 Feb 16 2025 08:33:16

%S 1,1,-1,0,0,-1,0,-1,0,0,0,1,0,-1,0,1,-1,0,-1,0,0,0,0,-2,1,0,2,1,0,1,0,

%T 0,0,1,1,1,0,0,0,0,-1,0,0,0,-1,0,1,0,-1,0,0,-2,0,0,0,-1,-1,-1,0,0,1,0,

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

%N Expansion of f(x) * f(x^11) in powers of x where f() is a Ramanujan theta function.

%C Number 69 of the 74 eta-quotients listed in Table I of Martin (1996).

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

%H G. C. Greubel, <a href="/A208664/b208664.txt">Table of n, a(n) for n = 0..1000</a>

%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>

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

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

%F Expansion of q^(-1/2) * eta(q^2)^3 * eta(q^22)^3 / (eta(q) * eta(q^4) * eta(q^11) * eta(q^44)) in powers of q.

%F Euler transform of period 44 sequence [ 1, -2, 1, -1, 1, -2, 1, -1, 1, -2, 2, -1, 1, -2, 1, -1, 1, -2, 1, -1, 1, -4, 1, -1, 1, -2, 1, -1, 1, -2, 1, -1, 2, -2, 1, -1, 1, -2, 1, -1, 1, -2, 1, -2, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (176 t)) = 176^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

%F a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(11^e) = (-1)^e, b(p^e) = s * ((e-1)%3 - 1) if f=0, b(p^e) = s * (e + 1) if f=3, b(p^e) = s * (1 + (-1)^e) / 2 if f=1 where s = (-1)^(e*(p-1)/2) and f = number of zeros of x^3-x^2-x-1 modulo p.

%F a(n) = (-1)^n * A030200(n). a(11*n + 3) = a(11*n + 6) = a(11*n + 8) = a(11*n + 9) = a(11*n + 10) = 0. a(11*n + 5) = -a(n).

%e G.f. = 1 + x - x^2 - x^5 - x^7 + x^11 - x^13 + x^15 - x^16 - x^18 - 2*x^23 + ...

%e G.f. = q + q^3 - q^5 - q^11 - q^15 + q^23 - q^27 + q^31 - q^33 - q^37 - 2*q^47 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^11] , {x, 0, n}]; (* _Michael Somos_, Jun 09 2015 *)

%o (PARI) {a(n) = my(A, p, e, f); 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, p==11, (-1)^e, f = sum( k=0, p-1, (k^3-k^2-k-1)%p == 0); (-1)^(e*(p-1)/2) * if( f==0, (e-1)%3-1, f==1, (1 + (-1)^e) / 2, e+1))))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^22 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^11 + A) * eta(x^44 + A)), n))};

%Y Cf. A030200.

%K sign,changed

%O 0,24

%A _Michael Somos_, Mar 05 2012