A258292 - OEIS

%I #10 Sep 08 2022 08:46:12

%S 1,-2,1,0,-2,2,0,0,1,4,-4,0,0,-4,0,0,-2,2,4,0,2,0,0,0,0,-6,2,0,0,2,0,

%T 0,1,0,-4,0,4,-4,0,0,-4,2,0,0,0,8,0,0,0,-2,3,0,-4,2,0,0,0,0,-4,0,0,-4,

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

%N Expansion of psi(-q)^2 * chi(q^3)^2 in powers of q where psi(), f() 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="/A258292/b258292.txt">Table of n, a(n) for n = 0..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 f(q) * psi(-q)^2 / psi(-q^3) in powers of q where psi(), f() are Ramanujan theta functions.

%F Expansion of f(x*w, x/w)^2 in powers of x where w is a primitive cube root of unity and f() is Ramanujan's general theta function.

%F Expansion of (eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)))^2 in powers of q.

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

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A122856.

%F G.f.: (Product_{k>0} (1 + x^k) * (1 + x^(3*k)) / (1 - x^(2*k) + x^(4*k)))^2.

%F a(n) = (-1)^n * A258279(n).  Convolution square of A089807.

%F a(2*n) = A258228(n). a(3*n + 1) = -2 * A122865(n). a(3*n + 2) = A122856(n). a(4*n) = a(n). a(4*n + 3) = 0. a(12*n + 1) = -2 * A002175(n).

%F a(18*n) = A004018(n). a(18*n + 3) = a(18*n + 6) = a(18*n + 12) = 0.

%e G.f. = 1 - 2*q + q^2 - 2*q^4 + 2*q^5 + q^8 + 4*q^9 - 4*q^10 - 4*q^13 + ...kkj

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/3, q]^2, {q, 0, n}];

%t a[ n_] := SeriesCoefficient[ q^(1/8) QPochhammer[ -q^3] EllipticTheta[ 2, Pi/4, q^(1/2)]^2 / (Sqrt[2] EllipticTheta[ 2, Pi/4, q^(3/2)]), {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)))^2, n))};

%o (Magma) A := Basis( ModularForms( Gamma1(36), 1), 82); A[1] - 2*A[2] + A[3] - 2*A[5] + 2*A[6] + A[9] + 4*A[10] - 4*A[11] - 4*A[14] - 2*A[17] + 2*A[18

%o ] + 4*A[19];

%Y Cf. A002175, A004018, A089807, A122856, A122865, A258228, A258279.

%K sign

%O 0,2

%A _Michael Somos_, May 25 2015