A134077 - OEIS

%I #15 Mar 12 2021 22:24:45

%S 1,-5,6,8,-23,12,14,-30,18,20,-40,24,31,-77,30,32,-60,48,38,-70,42,44,

%T -138,48,57,-90,54,72,-100,60,62,-184,84,68,-120,72,74,-155,96,80,

%U -239,84,108,-150,90,112,-160,120,98,-276,102,104,-240,108,110,-190,114

%N Expansion of psi(x) * phi(-x)^3 / chi(-x^3)^3 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

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

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H G. C. Greubel, <a href="/A134077/b134077.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 psi(x)^4 - 9 * x * psi(x^3)^4 in powers of x where psi() is a Ramanujan theta function.

%F Expansion of x^(-1/2) * (b(x)^3 * c(x^2)^2 / (3 * c(x)))^(1/2) in powers of x where b(), c() are cubic AGM functions.

%F Expansion of q^(-1/2) * eta(q)^5 * eta(q^6)^3 / (eta(q^2) * eta(q^3)^3) in powers of q.

%F Euler transform of period 6 sequence [-5, -4, -2, -4, -5, -4, ...].

%F a(n) = b(2*n+1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1)/(p - 1) if p>5.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 18 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A124449

%F G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(2*k))^2 * (1 - x^k + x^(2*k))^3.

%F G.f.: Sum_{k>0} k * f(x^k) - 9 * k * f(x^(3*k)) where f(x) = x * (1 - x) / ((1 + x) * (1 + x^2)).

%F G.f.: f(x) - 3 * f(x^2) - 9 * f(x^3) + 2 * f(x^4) + 27 * f(x^6) - 18 * f(x^12) where f() is the g.f. of A000203.

%F a(n) = A131944(2*n + 1) = A118271(2*n + 1). a(3*n + 2) = 6 * A098098(n).

%e G.f. = 1 - 5*x + 6*x^2 + 8*x^3 - 23*x^4 + 12*x^5 + 14*x^6 - 30*x^7 + 18*x^8 + ...

%e G.f. = q - 5*q^3 + 6*q^5 + 8*q^7 - 23*q^9 + 12*q^11 + 14*q^13 - 30*q^15 + ...

%t a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 4, 0, x]^3 QPochhammer[ -x^3, x^3]^3, {x, 0, n}]; (* _Michael Somos_, Oct 27 2015 *)

%t a[ n_] := SeriesCoefficient[ (1/16) x^(-1/2) (EllipticTheta[ 2, 0, x^(1/2)]^4 - 9 EllipticTheta[ 2, 0, x^(3/2)]^4), {x, 0, n}]; (* _Michael Somos_, Oct 27 2015 *)

%o (PARI) {a(n) = my(A); if ( n<0, 0, A = x * O(x^n) ; polcoeff( eta(x + A)^5 * eta(x^6 + A)^3 / ( eta(x^2 + A) * eta(x^3 + A)^3 ), n))};

%Y Cf. A000203, A098098, A118271, A124449, A131944.

%K sign

%O 0,2

%A _Michael Somos_, Oct 06 2007