A232635 - OEIS

%I #10 Mar 12 2021 22:24:47

%S 1,1,4,5,4,8,1,4,8,4,13,12,4,8,8,1,12,8,8,12,16,13,4,16,4,12,20,8,5,

%T 12,12,12,16,8,8,20,17,16,16,8,20,24,8,8,16,1,24,16,8,12,20,16,12,28,

%U 12,33,20,8,16,12,16,20,24,8,16,32,1,12,32,8,16,32,8

%N Expansion of psi(x) * phi(x^2)^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).

%C Theta function of cubic lattice with respect to point (1/4, 0, 0).

%H G. C. Greubel, <a href="/A232635/b232635.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^4)^10 / (eta(q) * eta(q^2)^2 * eta(q^8)^4) in powers of q.

%F Euler transform of period 8 sequence [ 1, 3, 1, -7, 1, 3, 1, -3, ...].

%F a(3*n + 2) = 4 * A213023(n).

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

%e G.f. = q + q^9 + 4*q^17 + 5*q^25 + 4*q^33 + 8*q^41 + q^49 + 4*q^57 + 8*q^65 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q^4]^10 / (QPochhammer[ q] QPochhammer[ q^2]^2 QPochhammer[ q^8]^4), {q, 0, n}]

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

%Y Cf. A213023.

%K nonn

%O 0,3

%A _Michael Somos_, Nov 27 2013