A246811 - OEIS

%I #10 Feb 16 2025 08:33:23

%S 1,4,4,0,5,12,4,0,8,12,8,0,5,16,12,0,8,24,4,0,16,12,12,0,9,24,12,0,8,

%T 36,12,0,16,12,16,0,8,28,16,0,17,36,8,0,24,24,8,0,8,36,28,0,16,36,12,

%U 0,16,24,20,0,13,24,24,0,24,60,8,0,16,36,16,0,16,28

%N Expansion of phi(x)^2 * psi(x^4) in powers of x where phi(), psi() 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="/A246811/b246811.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="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of psi(x)^4 / phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

%F Expansion of q^(-1/2) * eta(q^2)^10 * eta(q^8)^2 / (eta(q)^4 * eta(q^4)^5) in powers of q.

%F Euler transform of period 8 sequence [4, -6, 4, -1, 4, -6, 4, -3, ...].

%F 2 * a(n) = A033717(4*n + 2). a(2*n) = A045834(n). a(4*n) = A213022(n). a(4*n + 1) = 4 * A008443(n). a(4*n + 2) = 4 * A045831(n). a(4*n + 3) = 0.

%e G.f. = 1 + 4*x + 4*x^2 + 5*x^4 + 12*x^5 + 4*x^6 + 8*x^8 + 12*x^9 + ...

%e G.f. = q + 4*q^3 + 4*q^5 + 5*q^9 + 12*q^11 + 4*q^13 + 8*q^17 + 12*q^19 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^2 EllipticTheta[ 2, 0, x^2] / (2 x^(1/2)), {x, 0, n}];

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)]^4 / (16 x^(1/2) EllipticTheta[ 3, 0, x^2]), {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^8 + A)^2 / (eta(x + A)^4 * eta(x^4 + A)^5), n))};

%Y Cf. A008443, A033717, A045831, A045834, A213022.

%K nonn,changed

%O 0,2

%A _Michael Somos_, Sep 03 2014