cp's OEIS Frontend

A217786 Expansion of (psi(x^3) / psi(x))^2 in powers of x where psi() is a Ramanujan theta function.

%I A217786 #22 Feb 16 2025 08:33:18
%S A217786 1,-2,3,-4,7,-12,17,-24,36,-52,71,-96,133,-182,240,-316,420,-552,713,
%T A217786 -916,1182,-1516,1920,-2424,3063,-3852,4806,-5976,7430,-9204,11336,
%U A217786 -13924,17088,-20908,25473,-30960,37586,-45518,54939,-66172,79603,-95556,114399
%N A217786 Expansion of (psi(x^3) / psi(x))^2 in powers of x where psi() is a Ramanujan theta function.
%C A217786 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A217786 G. C. Greubel, <a href="/A217786/b217786.txt">Table of n, a(n) for n = 0..1000</a>
%H A217786 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A217786 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A217786 Expansion of q^(-1/2) * eta(q)^2 * eta(q^6)^4 / (eta(q^2)^4 * eta(q^3)^2) in powers of q.
%F A217786 Euler transform of period 6 sequence [ -2, 2, 0, 2, -2, 0, ...].
%F A217786 Given g.f. A(x), B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (1 - v) * (v - u^2) - 4 * u^2 * v.
%F A217786 Given g.f. A(x), B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = u * (u + 3*v)^2 - v * (1 + 3*u*v)^2.
%F A217786 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A217771.
%F A217786 G.f.: Product_{k>0} (1 + x^k + x^(2*k))^2 * (1 - x^k + x^(2*k))^4.
%F A217786 Convolution square of A101195. Convolution inverse of A058487.
%F A217786 a(n) = - A139216(6*n + 3). - _Michael Somos_, Sep 07 2015
%F A217786 a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (6^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 06 2018
%e A217786 G.f. = 1 - 2*x + 3*x^2 - 4*x^3 + 7*x^4 - 12*x^5 + 17*x^6 - 24*x^7 + 36*x^8 + ...
%e A217786 G.f. = q - 2*q^3 + 3*q^5 - 4*q^7 + 7*q^9 - 12*q^11 + 17*q^13 - 24*q^15 + 36*q^17 + ...
%t A217786 a[ n_] := SeriesCoefficient[ x^(-1/2) (EllipticTheta[ 2, 0, x^(3/2)] / EllipticTheta[ 2, 0, x^(1/2)])^2, {x, 0, n}];
%o A217786 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^4 / (eta(x^2 + A)^4 * eta(x^3 + A)^2), n))};
%Y A217786 Cf. A058487, A101195, A139216, A217771.
%K A217786 sign
%O A217786 0,2
%A A217786 _Michael Somos_, Mar 24 2013