A213266 - OEIS

%I #14 Mar 12 2021 22:24:46

%S 1,1,-1,-1,0,1,0,-1,1,2,0,-3,0,2,0,-3,0,5,0,-4,-2,4,0,-5,0,7,2,-7,0,5,

%T 0,-10,1,12,0,-10,0,14,-4,-17,0,21,0,-22,4,24,0,-34,0,33,1,-36,0,45,0,

%U -45,-8,52,0,-55,0,62,8,-71,0,70,0,-88,2,96,0,-98

%N Expansion of psi(q) * psi(q^9) / (psi(q^2) * chi(q^3) * psi(-q^9)) 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).

%H G. C. Greubel, <a href="/A213266/b213266.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^2)^3 * eta(q^3) * eta(q^12) * eta(q^18)^3 / (eta(q) * eta(q^4)^2 * eta(q^6)^2 * eta(q^9)^2 * eta(q^36)) in powers of q.

%F Euler transform of period 36 sequence [ 1, -2, 0, 0, 1, -1, 1, 0, 2, -2, 1, 0, 1, -2, 0, 0, 1, -2, 1, 0, 0, -2, 1, 0, 1, -2, 2, 0, 1, -1, 1, 0, 0, -2, 1, 0, ...].

%F a(n) = A182038(n) unless n=0. a(6*n) = 0 unless n=0. a(6*n + 4) = 0. a(6*n + 2) = -A092848(n).

%e 1 + q - q^2 - q^3 + q^5 - q^7 + q^8 + 2*q^9 - 3*q^11 + 2*q^13 - 3*q^15 + ...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q^2]^3* eta[q^3]*eta[q^12]*eta[q^18]^3/(eta[q]*eta[q^4]^2*eta[q^6]^2*eta[q^9]^2*eta[q^36]), {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Mar 19 2018 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A) * eta(x^18 + A)^3 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^2 * eta(x^9 + A)^2 * eta(x^36 + A)), n))}

%o (PARI) q='q+O('q^99); Vec(eta(q^2)^3*eta(q^3)*eta(q^12)*eta(q^18)^3 /(eta(q)*eta(q^4)^2*eta(q^6)^2*eta(q^9)^2*eta(q^36))) \\ _Altug Alkan_, Mar 20 2018

%Y Cf. A092848, A182038.

%K sign

%O 0,10

%A _Michael Somos_, Jun 07 2012