|
%I #9 Mar 12 2021 22:24:48
%S 1,-1,1,1,0,-1,-1,-1,-1,0,-1,0,1,1,-1,0,1,1,0,1,0,0,0,-1,1,1,0,-1,0,0,
%T 0,0,-1,0,0,2,2,-1,-1,-1,-1,2,-1,1,0,-1,0,0,-1,0,0,-1,-1,0,-1,0,0,0,0,
%U -1,0,0,1,0,0,0,0,1,0,0,1,1,-1,-1,2,-1,0,0,2
%N Expansion of f(-x^5) * f(-x^7) - x * f(-x) * f(-x^35) 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="/A277582/b277582.txt">Table of n, a(n) for n = 0..2500</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) * (eta(q^10) * eta(q^14) - eta(q^2) * eta(q^70)) in powers of q^2.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (140 t)) = 140^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
%e G.f. = 1 - x + x^2 + x^3 - x^5 - x^6 - x^7 - x^8 - x^10 + x^12 + x^13 + ...
%e G.f. = q - q^3 + q^5 + q^7 - q^11 - q^13 - q^15 - q^17 - q^21 + q^25 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^5] QPochhammer[ x^7] - x QPochhammer[ x] QPochhammer[ x^35], {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A) * eta(x^7 + A) - x * eta(x + A) * eta(x^35 + A), n))};
%o (PARI) {a(n) = my(j=24*n+12, n1, n2, t1, t2); if( n<0, 0, n1 = sqrtint(j\7); n2 = sqrtint(j\35); sum(k = (-n1+4)\6, (n1-1)\6, if( (t1 = j-7*(6*k+1)^2)%5 ==0 && issquare( t1/5, &t2), if( t2%6 == 5, t2=-t2); m = (t2-1)/6; (-1)^(k+m))) - sum(k = (-n2+4)\6, (n2-1)\6, t1 = j-35*(6*k+1)^2; if( issquare( t1, &t2), if( t2%6 == 5, t2=-t2); m = (t2-1)/6; (-1)^(k+m))))};
%Y Cf. A030216.
%K sign
%O 0,36
%A _Michael Somos_, Oct 21 2016
|
