cp's OEIS Frontend

A256269 Expansion of psi(-q) * chi(q^3) * phi(q^9) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

%I A256269 #18 Feb 16 2025 08:33:25
%S A256269 1,-1,0,0,-1,0,0,0,0,4,-2,0,0,-2,0,0,-1,0,4,0,0,0,0,0,0,-3,0,0,0,0,0,
%T A256269 0,0,0,-2,0,4,-2,0,0,-2,0,0,0,0,8,0,0,0,-1,0,0,-2,0,0,0,0,0,-2,0,0,-2,
%U A256269 0,0,-1,0,0,0,0,0,0,0,4,-2,0,0,0,0,0,0,0,4
%N A256269 Expansion of psi(-q) * chi(q^3) * phi(q^9) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
%C A256269 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A256269 G. C. Greubel, <a href="/A256269/b256269.txt">Table of n, a(n) for n = 0..1000</a>
%H A256269 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A256269 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A256269 Expansion of eta(q) * eta(q^4) * eta(q^6)^2 * eta(q^18)^5 / (eta(q^2) * eta(q^3) * eta(q^12) * eta(q^9)^2 * eta(q^36)^2) in powers of q.
%F A256269 Euler transform of period 36 sequence [ -1, 0, 0, -1, -1, -1, -1, -1, 2, 0, -1, -1, -1, 0, 0, -1, -1, -4, -1, -1, 0, 0, -1, -1, -1, 0, 2, -1, -1, -1, -1, -1, 0, 0, -1, -2, ...].
%F A256269 a(3*n + 1) = - A122865(n). a(6*n + 4) = - A122856(n). a(3*n + 2) = a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.
%e A256269 G.f. = 1 - q - q^4 + 4*q^9 - 2*q^10 - 2*q^13 - q^16 + 4*q^18 - 3*q^25 + ...
%t A256269 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)] / (2^(1/2) q^(1/8)) QPochhammer[ -q^3, q^6] EllipticTheta[ 3, 0, q^9], {q, 0, n}];
%o A256269 (PARI) {a(n) = if( n<1, n==0, (-1)^(n%3) * (n%3<2) * sumdiv(n, d, [ 0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};
%o A256269 (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^18 + A)^5 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) * eta(x^9 + A)^2 * eta(x^36 + A)^2), n))};
%o A256269 (Magma) A := Basis( ModularForms( Gamma1(36), 1), 82); A[1] - A[2] - A[5] + 4*A[10] - 2*A[11] - 2*A[14] - A[17] + 4*A[19];
%Y A256269 Cf. A122856, A122865.
%K A256269 sign
%O A256269 0,10
%A A256269 _Michael Somos_, Jun 01 2015