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A262726 Expansion of phi(-x) * psi(x^6) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A262726 #15 Feb 16 2025 08:33:27
%S A262726 1,-2,0,0,2,0,1,-2,0,-2,2,0,0,0,0,-2,2,0,1,-2,0,0,4,0,0,-2,0,-2,0,0,0,
%T A262726 -2,0,0,2,0,3,-2,0,0,2,0,2,-2,0,-2,0,0,0,-2,0,0,2,0,2,-2,0,0,0,0,1,-4,
%U A262726 0,0,4,0,0,-2,0,-2,2,0,2,0,0,0,2,0,0,0,0
%N A262726 Expansion of phi(-x) * psi(x^6) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A262726 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A262726 G. C. Greubel, <a href="/A262726/b262726.txt">Table of n, a(n) for n = 0..10000</a>
%H A262726 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A262726 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A262726 Expansion of q^(-3/4) * eta(q)^2 * eta(q^12)^2 / (eta(q^2) * eta(q^6)) in powers of q.
%F A262726 Euler transform of period 12 sequence [-2, -1, -2, -1, -2, 0, -2, -1, -2, -1, -2, -2, ...].
%F A262726 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 A262726 a(n) = (-1)^n * A112605(n). -2 * a(n) = A138270(2*n + 1).
%F A262726 a(2*n) = A112608(n). a(2*n + 1) = -2 * A112609(n). a(3*n + 2) = 0.
%F A262726 a(n) = A262780(2*n + 1). - _Michael Somos_, Oct 01 2015
%e A262726 G.f. = 1 - 2*x + 2*x^4 + x^6 - 2*x^7 - 2*x^9 + 2*x^10 - 2*x^15 + 2*x^16 + ...
%e A262726 G.f. = q^3 - 2*q^7 + 2*q^19 + q^27 - 2*q^31 - 2*q^39 + 2*q^43 - 2*q^63 + ...
%t A262726 a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 4 n + 3, KroneckerSymbol[ -3, #] &]];
%t A262726 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^3] / (2 x^(3/4)), {x, 0, n}];
%t A262726 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 + 3))]; (* _Michael Somos_, Oct 01 2015 *)
%o A262726 (PARI) {a(n) = if( n<0, 0, (-1)^n * sumdiv(4*n + 3, d, kronecker(-3, d)))};
%o A262726 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^12 + A)^2 / (eta(x^2 + A) * eta(x^6 + A)), n))};
%o A262726 (PARI) {a(n) = my(A, p, e); if( n<0, 0, A = factor(4*n + 3); 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 )))}; /* _Michael Somos_, Oct 01 2015 */
%Y A262726 Cf. A112605, A112608, A112609, A138270, A262780.
%K A262726 sign
%O A262726 0,2
%A A262726 _Michael Somos_, Sep 28 2015