A263452 - OEIS

%I #11 Mar 12 2021 22:24:48

%S 1,1,2,0,2,1,2,0,1,2,2,0,3,1,4,0,5,3,2,0,3,3,4,0,4,2,4,0,3,2,4,0,4,2,

%T 4,0,5,5,4,0,3,3,8,0,7,3,6,0,4,4,4,0,6,4,4,0,9,3,6,0,4,4,4,0,4,3,8,0,

%U 5,5,6,0,9,3,4,0,7,6,6,0,7,6,10,0,6,3

%N Expansion of f(-q^3)^3 * psi(q^12) / f(-q) in powers of q where ps(), f() 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="/A263452/b263452.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="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q^(-11/6) * eta(q^3)^3 * eta(q^24)^2 / (eta(q) * eta(q^12)) in powers of q.

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

%F 2 * a(n) = A261444(2*n + 1).  a(4*n + 1) = A212907(n). a(4*n + 3) = 0.

%F -2 * a(n) = A263527(2*n + 3). - _Michael Somos_, Nov 05 2015

%e G.f. = 1 + x + 2*x^2 + 2*x^4 + x^5 + 2*x^6 + x^8 + 2*x^9 + 2*x^10 + ...

%e G.f. = q^11 + q^17 + 2*q^23 + 2*q^35 + q^41 + 2*q^47 + q^59 + 2*q^65 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q^3]^3 EllipticTheta[ 2, 0, q^6] / ( 2 q^(3/2) QPochhammer[ q]), {q, 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^24 + A)^2 / (eta(x + A) * eta(x^12 + A)), n))};

%Y Cf. A212907, A261444, A263527.

%K nonn

%O 0,3

%A _Michael Somos_, Oct 18 2015