A260109 - OEIS

%I #13 Aug 02 2018 02:32:46

%S 1,3,4,6,9,12,14,12,16,18,18,24,21,27,28,30,36,24,38,42,40,42,36,48,

%T 43,48,52,48,54,60,62,54,56,66,72,72,74,63,72,78,81,84,64,84,88,84,90,

%U 72,98,108,100,102,72,108,110,114,112,96,126,96,133,120,104

%N Expansion of f(x^3) * f(-x^3)^2 * psi(x)^2 / psi(-x) in powers of x where psi(), 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="/A260109/b260109.txt">Table of n, a(n) for n = 0..2500</a>

%F Expansion of psi(-x^3) * phi(-x^6)^2 * psi(x)^2 / psi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.

%F Expansion of phi(x) * f(x, x^5) * f(x^2, x^4)^2 in powers of x. - _Michael Somos_, Jul 18 2015

%F Expansion of q^(-1/2) * eta(q^2)^5 * eta(q^3) * eta(q^6)^3 / (eta(q)^3 * eta(q^4) * eta(q^12)) in powers of q.

%F Euler transform of period 12 sequence [ 3, -2, 2, -1, 3, -6, 3, -1, 2, -2, 3, -4, ...].

%F a(n) = A124815(2*n + 1). a(3*n + 1) = 3 * a(n).

%e G.f. = 1 + 3*x + 4*x^2 + 6*x^3 + 9*x^4 + 12*x^5 + 14*x^6 + 12*x^7 + ...

%e G.f. = q + 3*q^3 + 4*q^5 + 6*q^7 + 9*q^9 + 12*q^11 + 14*q^13 + 12*q^15 + ...

%t a[ n_] := SeriesCoefficient[ 1/4 x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)]^2 EllipticTheta[ 2, Pi/4, x^(3/2)] EllipticTheta[ 4, 0, x^6]^2 / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}];

%t a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-1/8) QPochhammer[ -x^3] QPochhammer[ x^3]^2 EllipticTheta[ 2, 0, x^(1/2)]^2 / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A) * eta(x^6 + A)^3 / (eta(x + A)^3 * eta(x^4 + A) * eta(x^12 + A)), n))};

%Y Cf. A124815.

%K nonn

%O 0,2

%A _Michael Somos_, Jul 16 2015