A225853 - OEIS

%I #14 Mar 12 2021 22:24:47

%S 1,2,0,0,3,2,0,0,4,6,0,0,7,8,0,0,13,14,0,0,19,20,0,0,29,34,0,0,43,46,

%T 0,0,62,70,0,0,90,96,0,0,126,138,0,0,174,186,0,0,239,262,0,0,325,346,

%U 0,0,435,472,0,0,580,620,0,0,769,826,0,0,1007,1072,0

%N Expansion of phi(x) / f(-x^4) in powers of x where phi(), f() 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="/A225853/b225853.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 chi(x)^2 * chi(-x^2) = chi(x)^3 * chi(-x) = chi(-x^2)^3 / chi(-x)^2 in powers of x where chi() is a Ramanujan theta function.

%F Expansion of q^(1/4) * eta(q^2)^5 / (eta(q)^2 * eta(q^4)^3) in powers of q.

%F Euler transform of period 4 sequence [ 2, -3, 2, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A029552.

%F G.f.: Product_{k>0} (1 - x^(4*k-2))^3 / (1 - x^(2*k-1))^2 = (Sum_{k in Z} x^k^2) / (Product_{k>0} (1 - x^(4*k))).

%F a(n) = (-1)^n * A143161(n). a(4*n + 2) = a(4*n + 3) = 0.

%e 1 + 2*x + 3*x^4 + 2*x^5 + 4*x^8 + 6*x^9 + 7*x^12 + 8*x^13 + 13*x^16 + ...

%e 1/q + 2*q^5 + 3*q^23 + 2*q^29 + 4*q^47 + 6*q^53 + 7*q^71 + 8*q^77 + 13*q^95 + ...

%t a[n_]:= SeriesCoefficient[EllipticTheta[3,0,q]/QPochhammer[q^4],{q,0,n}];

%t a[n_]:= SeriesCoefficient[QPochhammer[q^2,q^4]^3/QPochhammer[q,q^2]^2, {q,0,n}];

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

%Y Cf. A029552, A143161.

%K nonn

%O 0,2

%A _Michael Somos_, May 17 2013