A124452 - OEIS

%I #16 Mar 12 2021 22:24:44

%S 1,-1,-1,1,-1,0,1,0,-1,3,0,-2,1,0,0,0,-1,-2,3,-2,0,0,-2,0,1,-1,0,5,0,

%T 0,0,0,-1,2,-2,0,3,0,-2,0,0,-2,0,-2,-2,0,0,0,1,-1,-1,2,0,0,5,0,0,2,0,

%U -2,0,0,0,0,-1,0,2,-2,-2,0,0,0,3,-2,0,1,-2,0,0,0,0,7,-2,-2,0,0,-2,0,-2,-2,0,0,0,0,0,0,1,-2,-1,6,-1,0,2,0,0

%N Expansion of psi(-q) * psi(-q^2) * chi(q^3) * chi(q^6) in powers of q where psi(), chi() are Ramanujan theta functions.

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

%C a(n) is nonzero if and only if n is in A002479.

%H G. C. Greubel, <a href="/A124452/b124452.txt">Table of n, a(n) for n = 0..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) * eta(q^6) * eta(q^8) * eta(q^12) / (eta(q^3) * eta(q^24)) in powers of q.

%F a(n) = -b(n) where b(n) is multiplicative with b(2^e) = 1, b(3^e) = 1 - 2*e, b(p^e) = 1+e if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8).

%F Euler transform of period 24 sequence [ -1, -1, 0, -1, -1, -1, -1, -2, 0, -1, -1, -2, -1, -1, 0, -2, -1, -1, -1, -1, 0, -1, -1, -2, ...].

%F Moebius transform is period 24 sequence [ -1, 0, 2, 0, 1, 0, 1, 0, 2, 0, -1, 0, 1, 0, -2, 0, -1, 0, -1, 0, -2, 0, 1, 0, ...].

%F a(8*n + 5) = a(8*n + 7) = 0. a(2*n) = a(n). a(3*n) >= 0.

%e 1 - q - q^2 + q^3 - q^4 + q^6 - q^8 + 3*q^9 - 2*q^11 + q^12 - q^16 - 2*q^17 + ...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]* eta[q^6]*eta[q^8]*eta[q^12]/(eta[q^3]*eta[q^24]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Mar 08 2018 *)

%o (PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); - prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 1, if( p==3, 1 - 2*e, if( p%8<4, e+1, !(e%2)))))))}

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A) * eta(x^8 + A) * eta(x^12 + A) / (eta(x^3 + A) * eta(x^24 + A)), n))}

%o (PARI) q='q+O('q^99); Vec(eta(q)*eta(q^6)*eta(q^8)*eta(q^12)/(eta(q^3)*eta(q^24))) \\ _Altug Alkan_, Mar 09 2018

%Y Cf. A002479.

%K sign

%O 0,10

%A _Michael Somos_, Nov 02 2006