A128765 - OEIS

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

%S 1,1,-1,0,1,-1,-2,0,2,0,0,2,1,-1,-2,-2,0,1,-2,0,6,3,-5,-2,5,-3,-10,2,

%T 8,-1,-2,6,6,-2,-8,-6,2,2,-8,-2,21,11,-18,-4,18,-11,-32,4,26,-1,-10,

%U 18,20,-8,-26,-18,10,8,-26,-2,61,27,-53,-12,52,-26,-88,12,74,-6,-32,42,58,-17,-74,-40,34,16,-74,-8,156,66,-136,-26

%N Expansion of psi(q) * psi(q^10) / ( psi(q^2) * psi(q^5)) in powers of q where psi() is a Ramanujan theta function.

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

%H G. C. Greubel, <a href="/A128765/b128765.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 q^(-1/2) * eta(q^2)^3 * eta(q^5) * eta(q^20)^2 / (eta(q) * eta(q^4)^2 * eta(q^10)^3) in powers of q.

%F Euler transform of period 20 sequence [ 1, -2, 1, 0, 0, -2, 1, 0, 1, 0, 1, 0, 1, -2, 0, 0, 1, -2, 1, 0, ...].

%F Given g.f. A(x), then B(q) = q*A(q^2) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (u - v)^2.

%F G.f.: Product_{k>0} (1 + x^k) * (1 + x^(10*k))^2 / ( (1 + x^(2*k))^2* (1+x^(5*k)) ).

%e G.f. = 1 + x - x^2 + x^4 - x^5 - 2*x^6 +  2*x^8 + 2*x^11 + x^12 - x^13 + ...

%e G.f. = q + q^3 - q^5 + q^9 - q^11 - 2*q^13 + 2*q^17 + 2*q^23 + q^25 - q^27 + ...

%t a[ n_] := SeriesCoefficient[ x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^5] / (EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^(5/2)]), {x, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^5 + A) * eta(x^20 + A)^2 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^10 + A)^3), n))};

%K sign

%O 0,7

%A _Michael Somos_, Mar 25 2007