A255258 - OEIS

%I #10 Feb 16 2025 08:33:25

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

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

%U 0,0,0,0,0,2,0,0,0,0,0,0,2,2,0,0,2,0,0

%N Expansion of q^2 * phi(q) * psi(q^16) in powers of q 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="/A255258/b255258.txt">Table of n, a(n) for n = 2..2500</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of eta(q^2)^5 * eta(q^32)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^16)) in powers of q.

%F Euler transform of period 32 sequence [ 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -2, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A224609.

%F (-1)^n * a(n) = A227395(n).

%F a(4*n) = a(4*n + 1) = a(8*n + 7) = 0. a(4*n + 2) = A113411(n). a(8*n + 3) = 2 * A033761(n).

%e G.f. = q^2 + 2*q^3 + 2*q^6 + 2*q^11 + 3*q^18 + 2*q^19 + 2*q^22 + 4*q^27 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^8] / 2, {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^32 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^16 + A)), n))};

%o (Magma) A := Basis( ModularForms( Gamma1(32), 1), 89); A[3] + 2*A[4] + 2*A[7] + 2*A[12];

%Y Cf. A033761, A113411, A224609, A227395.

%K nonn,changed

%O 2,2

%A _Michael Somos_, Feb 19 2015