A261426 - OEIS

%I #10 Mar 12 2021 22:24:48

%S 1,1,2,0,2,1,4,2,5,2,6,2,6,0,4,4,7,2,4,0,6,1,8,4,4,4,10,2,8,2,12,4,8,

%T 5,6,0,14,2,8,2,11,6,6,4,8,2,8,4,8,6,14,0,6,0,12,6,15,4,14,2,14,4,8,8,

%U 12,7,14,0,12,2,16,10,8,4,10,6,14,0,16,4,16

%N Expansion of f(-x^3)^3 * phi(x^6) / f(-x) 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).

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

%H G. C. Greubel, <a href="/A261426/b261426.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 (1/3) * q^(-1/3) * c(q) * phi(q^6) in powers of q where phi() is a Ramanujan theta function and c() is a cubic AGM function. - _Michael Somos_, Sep 01 2015

%F Expansion of q^(-1/3) * eta(q^3)^3 * eta(q^12)^5 / (eta(q) * eta(q^6)^2 * eta(q^24)^2) in powers of q.

%F Euler transform of period 24 sequence [ 1, 1, -2, 1, 1, 0, 1, 1, -2, 1, 1, -5, 1, 1, -2, 1, 1, 0, 1, 1, -2, 1, 1, -3, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (128/3)^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261426.

%F 24 * a(n) = A261394(6*n + 2).

%F a(n) = A261444(2*n). _Michael Somos_, Sep 01 2015

%e G.f. = 1 + x + 2*x^2 + 2*x^4 + x^5 + 4*x^6 + 2*x^7 + 5*x^8 + 2*x^9 + ...

%e G.f. = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 4*q^19 + 2*q^22 + 5*q^25 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 EllipticTheta[ 3, 0, x^6] / QPochhammer[ x], {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^12 + A)^5 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^24 + A)^2), n))};

%Y Cf. A033687, A261394, A261426, A261444.

%K nonn

%O 0,3

%A _Michael Somos_, Aug 18 2015