A134015 - OEIS

%I #12 Mar 12 2021 22:24:45

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

%T 0,0,0,-2,2,0,0,-4,2,0,0,0,2,0,0,0,1,0,0,-4,2,0,0,0,0,0,0,0,2,0,0,-2,

%U 4,0,0,-4,0,0,0,-2,2,0,0,0,0,0,0,-4,1,0,0

%N Expansion of (1 - phi(-q) * phi(q^4)) / 2 in powers of q where phi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A134015/b134015.txt">Table of n, a(n) for n = 1..10000</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 Moebius transform is period 16 sequence [ 1, -1, -1, -2, 1, 1, -1, 0, 1, -1, -1, 2, 1, 1, -1, 0, ...].

%F a(n) is multiplicative with a(2) = 0, a(2^e) = -2 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4).

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

%F G.f.: x / (1 + x^2) + x^3 / (1 + x^6) - 2 * x^4 / (1 + x^8) + ...

%F a(n) = -(-1)^n * A113406(n). -2 * a(n) = A134014(n) unless n=0. a(4*n) = -2 * A002654(n). a(4*n + 1) = A008441(n).

%e G.f. = x - 2*x^4 + 2*x^5 - 2*x^8 + x^9 + 2*x^13 - 2*x^16 + 2*x^17 + ...

%t a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, x] EllipticTheta[ 3, 0, x^4]) / 2, {x, 0, n}]; (* _Michael Somos_, Oct 28 2015 *)

%t a[ n_] := If[ n < 1 || Mod[n, 4] > 1, 0, (Mod[n, 2] 3 - 2) DivisorSum[ n, KroneckerSymbol[ -4, #]&]]; (* _Michael Somos_, Oct 28 2015 *)

%o (PARI) {a(n) = if( n<1 || n%4>1, 0, (n%2*3 - 2) * sumdiv(n, d, kronecker(-4, d)))};

%o (PARI) {a(n) = -(-1)^n * if( n<1, 0, qfrep([1, 0; 0, 4], n)[n])};

%Y Cf. A002654, A008441, A113406, A134014.

%K sign,mult

%O 1,4

%A _Michael Somos_, Oct 02 2007