|
%I #12 Mar 12 2021 22:24:45
%S 1,1,-1,0,0,-1,1,0,-1,0,1,0,0,1,-2,1,2,-3,1,1,-2,3,0,-3,1,2,-2,0,2,-6,
%T 3,4,-7,3,2,-5,6,2,-8,3,5,-6,2,4,-12,7,10,-15,6,5,-13,12,4,-18,7,11,
%U -14,6,10,-24,14,20,-32,12,12,-29,24,9,-36,15,22,-30
%N Expansion of f(q) * f(q^15) / (f(-q^6) * f(-q^10)) in powers of q where f() 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="/A145727/b145727.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) * eta(q^30))^3 / (eta(q) * eta(q^4) * eta(q^6) * eta(q^10) * eta(q^15) * eta(q^60)) in powers of q.
%F Euler transform of a period 60 sequence.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = f(t) where q = exp(2 Pi i t).
%F a(n) = A145726(n) unless n=0. Convolution inverse of A145728.
%e G.f. = 1 + q - q^2 - q^5 + q^6 - q^8 + q^10 + q^13 - 2*q^14 + q^15 + 2*q^16 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -q] QPochhammer[ -q^15] / (QPochhammer[ q^6] QPochhammer[ q^10]), {q, 0, n}]; (* _Michael Somos_, Sep 05 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^30 + A))^3 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A) * eta(x^60 + A)), n))};
%Y Cf. A145726, A145728.
%K sign
%O 0,15
%A _Michael Somos_, Oct 23 2008
|
