A145702 - OEIS

%I #14 Jul 25 2024 02:39:33

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

%T 5,-4,4,-5,6,-6,5,-6,8,-7,6,-8,11,-10,8,-11,13,-11,10,-13,16,-15,14,

%U -17,20,-18,17,-20,24,-23,21,-25,31,-29,26,-32,37,-34,32

%N Expansion of chi(-x) * chi(x^5) in powers of x where chi() is a Ramanujan theta function.

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

%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^(1/4) * eta(q) * eta(q^10)^2 / eta(q^2) / eta(q^5) / eta(q^20) in powers of q.

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

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A145703.

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

%F a(n) = (-1)^n * A139632(n). a(2*n) = A139631(n). a(2*n + 1) = - A145703(n).

%F a(n) = -(-1)^floor(n/2) * A145704(n) = (-1)^floor((n + 1)/2) * A145705(n). - _Michael Somos_, Sep 06 2015

%e G.f. = 1 - x - x^3 + x^4 - x^7 + x^8 - x^9 + x^10 - x^11 + 2*x^12 - x^13 + ...

%e G.f. = 1/q - q^3 - q^11 + q^15 - q^27 + q^31 - q^35 + q^39 - q^43 + 2*q^47 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^5, x^10], {x, 0, n}]; (* _Michael Somos_, Sep 06 2015 *)

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

%Y Cf. A139631, A139632, A145703, A145704, A145705.

%K sign

%O 0,13

%A _Michael Somos_, Oct 17 2008