A298931 - OEIS

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

%S 1,0,0,1,1,0,2,1,0,0,2,0,3,0,0,2,2,0,4,1,0,0,2,0,4,0,0,4,1,0,6,2,0,0,

%T 2,0,5,0,0,3,3,0,6,1,0,0,4,0,6,0,0,4,5,0,4,3,0,0,2,0,8,0,0,4,3,0,6,3,

%U 0,0,4,0,9,0,0,6,4,0,6,2,0,0,4,0,6,0,0

%N Expansion of psi(x^4) * c(x^3) / (3*x) where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.

%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="/A298931/b298931.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^(-3/2) * eta(q^8)^2 * eta(q^9)^3 / (eta(q^3) * eta(q^4)) in powers of q.

%F Euler transform of a period 72 sequence.

%F A005872(2*n + 3) = 6*a(n). a(3*n) = A298932(n). a(3*n + 1) = A263452(n-1). a(3*n + 2) = a(4*n + 1) = 0.

%e G.f. = q^3 + q^9 + q^11 + 2*q^15 + q^17 + 2*q^23 + 3*q^27 + 2*q^33 + ...

%e G.f. = 1 + x^3 + x^4 + 2*x^6 + x^7 + 2*x^10 + 3*x^12 + 2*x^15 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^8]^2 QPochhammer[ x^9]^3 / (QPochhammer[ x^3] QPochhammer[ x^4]), {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^8 + A)^2 * eta(x^9 + A)^3 / (eta(x^3 + A) * eta(x^4 + A)), n))};

%Y Cf. A005872, A263452, A298932.

%K nonn

%O 0,7

%A _Michael Somos_, Jan 29 2018