A262774 - OEIS

%I #9 Mar 12 2021 22:24:48

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

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

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

%N Expansion of psi(x^2) * phi(-x^3) 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 G. C. Greubel, <a href="/A262774/b262774.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 Expansion of q^(-1/4) * eta(q^3)^2 * eta(q^4)^2 / (eta(q^2) * eta(q^6)) in powers of q.

%F Euler transform of period 12 sequence [ 0, 1, -2, -1, 0, 0, 0, -1, -2, 1, 0, -2, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = 192^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

%F a(n) = (-1)^n * A112604(n). a(2*n) = A112606(n). a(2*n + 1) = -2 * A112607(n-1). a(3*n + 1) = 0.

%F a(6*n) = A131961(n).  a(6*n + 2) = A112608(n). a(6*n + 3) = -2 * A131963(n). a(6*n + 5) = -2 * A112609(n).

%e G.f. = 1 + x^2 - 2*x^3 - 2*x^5 + x^6 - 2*x^9 + 3*x^12 + 2*x^14 - 2*x^15 + ...

%e G.f. = q + q^9 - 2*q^13 - 2*q^21 + q^25 - 2*q^37 + 3*q^49 + 2*q^57 + ...

%t a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 4 n + 1, KroneckerSymbol[ -3, #]&]];

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

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^2 QPochhammer[ x^4]^2 / ( QPochhammer[ x^2]  QPochhammer[ x^6]), {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 @ (4 n + 1))];

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

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

%o (PARI) {a(n) = my(A, p, e); if( n<0, 0, A = factor(4*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. A112604, A112604, A112606, A112607, A112608, A112609, A131961, A131963.

%K sign

%O 0,4

%A _Michael Somos_, Oct 01 2015