A124449 - OEIS

%I #18 Sep 12 2023 02:23:36

%S 1,-3,3,-3,6,-9,8,-3,9,-18,12,-9,14,-24,18,-3,18,-27,20,-18,24,-36,24,

%T -9,31,-42,27,-24,30,-54,32,-3,36,-54,48,-27,38,-60,42,-18,42,-72,44,

%U -36,54,-72,48,-9,57,-93,54,-42,54,-81,72,-24,60,-90,60,-54,62,-96,72,-3,84,-108,68,-54,72,-144,72,-27,74,-114

%N Expansion of (phi(-q^3)^4 - phi(-q)^4)/8 in powers of q where phi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) := Product_{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) := Product_{k>=0} (1+q^(2k+1)) (A000700).

%H G. C. Greubel, <a href="/A124449/b124449.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 eta(q)^3*eta(q^6)^5/(eta(q^2)^3*eta(q^3)) in powers of q.

%F Euler transform of period 6 sequence [ -3, 0, -2, 0, -3, -4, ...].

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

%F G.f.: x*Product_{k>0} (1+x^(3k))*(1-x^(6k))^4/(1+x^k)^3.

%F a(3*n) = 3*a(n). a(4*n) = a(2*n).

%F A121443(n) = -a(2*n)/3.

%F Dirichlet g.f.: (1 - 1/2^(s-2)) * (1 - 1/2^(s-1)) * (1 - 1/3^s) * zeta(s-1) * zeta(s). - _Amiram Eldar_, Sep 12 2023

%t a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, -q^3]^4 - EllipticTheta[ 3, 0, -q]^4)/8, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* _G. C. Greubel_, Dec 16 2017 *)

%o (PARI) {a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod( k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, -3, if(p==3, 3^e, (p^(e+1)-1)/(p-1))))))}

%o (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)^3*eta(x^6+A)^5/ eta(x^2+A)^3/eta(x^3+A), n))}

%Y Cf. A000122, A000700, A010054, A121373, A121443.

%K sign,easy,mult

%O 1,2

%A _Michael Somos_, Nov 01 2006