%I #12 Sep 08 2022 08:46:11

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

%T 0,0,2,0,0,2,4,0,0,0,0,0,0,0,1,6,0,2,4,0,0,0,0,2,0,0,2,0,0,1,8,0,0,4,

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

%N Expansion of q * phi(q) * chi(q^3) * psi(-q^9) in powers of q where phi(), psi(), chi() 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="/A256276/b256276.txt">Table of n, a(n) for n = 1..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 eta(q^2)^5 * eta(q^6)^2 * eta(q^9) * eta(q^36) / (eta(q)^2 * eta(q^4)^2 * eta(q^3) * eta(q^12) * eta(q^18)) in powers of q.

%F Euler transform of period 36 sequence [ 2, -3, 3, -1, 2, -4, 2, -1, 2, -3, 2, -1, 2, -3, 3, -1, 2, -4, 2, -1, 3, -3, 2, -1, 2, -3, 2, -1, 2, -4, 2, -1, 3, -3, 2, -2, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A256269.

%F a(3*n) = a(4*n + 3) = 0. a(3*n + 1) = A122865(n). a(3*n + 2) = 2 * A122856(n). a(4*n + 1) = a(n). a(4*n) = a(n). a(6*n + 2) = 2 * A122865(n). a(6*n + 4) = A122856(n).

%e G.f. = q + 2*q^2 + q^4 + 4*q^5 + 2*q^8 + 2*q^10 + 2*q^13 + q^16 + 4*q^17 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(9/2)] / (2^(1/2) q^(1/8)) QPochhammer[ -q^3, q^6] EllipticTheta[ 3, 0, q], {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^6 + A)^2 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^3 + A) * eta(x^12 + A) * eta(x^18 + A)), n))};

%o (Magma) A := Basis( ModularForms( Gamma1(36), 1), 89); A[2] + 2*A[3]

%o + A[5] + 4*A[6] + 2*A[9] + 2*A[11] + 2*A[14] + A[17] + 4*A[18];

%Y Cf. A122856, A122865, A256269.

%K nonn

%O 1,2

%A _Michael Somos_, Jun 02 2015