A213622 - OEIS

%I #11 Mar 12 2021 22:24:46

%S 1,3,4,7,8,4,9,8,4,16,9,8,20,8,8,11,8,12,20,20,8,15,16,12,20,16,8,24,

%T 21,8,20,8,16,28,24,8,17,32,12,36,16,8,24,16,24,19,20,20,32,16,12,28,

%U 16,20,44,27,12,36,24,16,28,24,16,28,32,12,25,32,12,48

%N Expansion of phi(x) * psi(x) * phi(x^2) 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="/A213622/b213622.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 q^(-1/8) * eta(q^2)^5 * eta(q^4)^3 / (eta(q)^3 * eta(q^8)^2), in powers of q.

%F Euler transform of period 8 sequence [ 3, -2, 3, -5, 3, -2, 3, -3, ...].

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

%e q + 3*q^9 + 4*q^17 + 7*q^25 + 8*q^33 + 4*q^41 + 9*q^49 + 8*q^57 + 4*q^65 + ...

%t a[ n_] := SeriesCoefficient[ 1/2 EllipticTheta[ 2, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^4], {q, 0, 2 n + 1/4}]; Table[a[n], {n, 0, 80}]

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^4 + A)^3 / (eta(x + A)^3 * eta(x^8 + A)^2), n))}

%K nonn

%O 0,2

%A _Michael Somos_, Jun 16 2012