A145723 - OEIS

%I #13 Mar 12 2021 22:24:45

%S 1,1,-1,0,0,-2,-1,0,0,0,1,0,0,0,0,0,-1,2,0,0,2,2,0,0,0,-4,-1,0,0,0,2,

%T 0,-1,0,0,0,-2,2,0,0,3,4,-2,0,0,-8,-3,0,0,0,5,0,-2,0,0,0,-3,4,0,0,6,8,

%U -2,0,0,-14,-4,0,0,0,8,0,-3,0,0,0,-6,8,0,0

%N Expansion of q^(-1) * f(q) * chi(-q^5) / f(-q^20) in powers of q where f(), 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="/A145723/b145723.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^2)^3 * eta(q^5) / (eta(q) * eta(q^4) * eta(q^10) * eta(q^20)) in powers of q.

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

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 20^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A145722.

%F a(4*n + 2) = a(5*n + 2) = a(5*n + 3) = 0.

%F a(4*n) = A138527(n). a(4*n + 1) = - A147699(n).

%F Convolution inverse of A145724.

%e G.f. = 1/q + 1 - q - 2*q^4 - q^5 + q^9 - q^15 + 2*q^16 + 2*q^19 + 2*q^20 + ...

%t a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ -q]  QPochhammer[ q^5, q^10] / QPochhammer[ q^20], {q, 0, n}]; (* _Michael Somos_, Sep 05 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^5 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A) * eta(x^20 + A)), n))};

%Y Cf. A138527, A145722, A145724, A147699.

%K sign

%O -1,6

%A _Michael Somos_, Nov 06 2008