A193426 - OEIS

%I #17 Mar 12 2021 22:24:46

%S 0,1,1,0,0,-1,0,1,1,0,0,1,0,2,0,0,0,-1,0,0,2,0,0,-1,0,2,1,0,0,0,0,1,0,

%T 0,0,1,0,2,2,0,0,-2,0,0,0,0,0,1,0,1,0,0,0,-1,0,2,2,0,0,0,0,2,2,0,0,0,

%U 0,0,0,0,0,-1,0,2,1,0,0,-2,0,0,1,0,0,2,0,2,0,0,0,0,0,0,2,0,0,-1,0,3,0,0,0,0,0,2,0

%N Expansion of (a(q^2) + a(q^3) - 2*a(q^6)) / 6 in powers of q where a() is a cubic AGM 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="/A193426/b193426.txt">Table of n, a(n) for n = 1..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 (b(q^6)^2 / b(q^3) - b(q^2)) / 3 = (c(q^6) / c(q^3)) * (c(q^3) + c(q^6)) / 3 = q^2 * psi(q) * psi(q^9)^2 / psi(q^3) in powers of q where b(), c() are cubic AGM functions and psi() is a Ramanujan theta function.

%F Expansion of eta(q^2)^2 * eta(q^3) * eta(q^18)^4 / (eta(q) * eta(q^6)^2 * eta(q^9)^2) in powers of q.

%F Euler transform of period 18 sequence [ 1, -1, 0, -1, 1, 0, 1, -1, 2, -1, 1, 0, 1, -1, 0, -1, 1, -2, ...].

%F Moebius transform is period 18 sequence [ 0, 1, 1, -1, 0, -3, 0, 1, 0, -1, 0, 3, 0, 1, -1, -1, 0, 0, ...].

%F a(3*n) = A093829(n). a(6*n) = -A093829(n). a(6*n + 2) = A033687(n). A(6*n + 3) = A033762(n). a(3*n + 1) = a(6*n + 5) = 0. a(4*n) = a(n).

%e G.f. = q^2 + q^3 - q^6 + q^8 + q^9 + q^12 + 2*q^14 - q^18 + 2*q^21 - q^24 + ...

%t a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^9]^2 / EllipticTheta[ 2, 0, q^3], {q, 0, 2 n}];

%o (PARI) {a(n) = my(A, p, e, q, f); if( n<1, 0, f = valuation( n, 2); q = n / 2^f; A = factor(q); if( q%6 == 1, f%2, (-1)^f) *  prod( k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p==3, 1, if( p%6==1, e+1, !(e%2))))))};

%o (PARI) {a(n) = my(A); if( n<2, 0, n = n-2; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)^4 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^9 + A)^2), n))};

%Y Cf. A033687, A033762, A093829.

%K sign

%O 1,14

%A _Michael Somos_, Jul 27 2011