A262780 - OEIS

%I #8 Mar 12 2021 22:24:48

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

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

%U 0,-2,2,0,-4,0,0,2,0,0,0,3,0,-2,0,0,-2,0,0

%N Expansion of phi(-x^6) * psi(x^4) + x * phi(-x^2) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.

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

%H Antti Karttunen, <a href="/A262780/b262780.txt">Table of n, a(n) for n = 0..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 a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 19 (mod 24), b(p^e) = (-1)^e * (e+1) if p == 7, 13 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).

%F a(2*n) = A262774(n). a(2*n + 1) = A262726(n).

%F abs(a(n)) = A033762(n).

%e G.f. = 1 + x - 2*x^3 + x^4 - 2*x^6 + 2*x^9 - 2*x^10 + x^12 + x^13 + ...

%e G.f. = q + q^3 - 2*q^7 + q^9 - 2*q^13 + 2*q^19 - 2*q^21 + q^25 + ...

%t a[ n_] := If[ n < 0, 0, {1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, 1}[[Mod[n, 12, 1]]] DivisorSum[ 2 n + 1, KroneckerSymbol[ -3, #] &]];

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

%t a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 5, Mod[#, 2], Mod[#, 6] == 5, 1 - Mod[#2, 2], True, (#2  + 1) KroneckerSymbol[ 6, #]^#2] & @@@ FactorInteger @ (2 n + 1))];

%o (PARI) {a(n) = my(A, p, e); if( n<0, 0, A = factor(2*n + 1); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p%2, p%6 == 1, (e+1) * if( p%24 == 1 || p%24 == 19, 1, (-1)^e), 1-e%2 )))};

%Y Cf. A033762, A262726, A262774.

%K sign

%O 0,4

%A _Michael Somos_, Oct 01 2015