cp's OEIS Frontend

A138520 Expansion of 1 - q * (psi(q^5) / psi(q))^2 in powers of q where psi() is a Ramanujan theta function.

%I A138520 #16 Feb 16 2025 08:33:08
%S A138520 1,-1,2,-3,6,-11,16,-24,38,-57,82,-117,168,-238,328,-448,614,-834,
%T A138520 1114,-1480,1966,-2592,3384,-4398,5704,-7361,9436,-12045,15344,-19470,
%U A138520 24576,-30922,38822,-48576,60548,-75259,93342,-115454,142360,-175104,214958,-263262
%N A138520 Expansion of 1 - q * (psi(q^5) / psi(q))^2 in powers of q where psi() is a Ramanujan theta function.
%C A138520 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A138520 G. C. Greubel, <a href="/A138520/b138520.txt">Table of n, a(n) for n = 0..1000</a>
%H A138520 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A138520 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A138520 Expansion of (phi(-q^5) / phi(-q))^2 * (chi^5(-q) / chi(-q^5)) in powers of q where phi(), chi() are Ramanujan theta functions. - _Michael Somos_, Sep 16 2015
%F A138520 Expansion of (eta(q^5) / eta(q^2))^3 * eta(q) / eta(q^10) in powers of q.
%F A138520 Euler transform of period 10 sequence [ -1, 2, -1, 2, -4, 2, -1, 2, -1, 0, ...].
%F A138520 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v)^2 - u * (1 - v) * (5*u - 4).
%F A138520 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (1 - u) * (5*u - 4) * v * (1 - v) * (5*v - 4).
%F A138520 G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (4/5) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138522.
%F A138520 G.f.: Product_{k>0} P(5,x^k)^2 / ((1 + x^k)^4 * P(10,x^k)) where P(n,x) is the n-th cyclotomic polynomial.
%F A138520 a(n) = - A138519(n) unless n=0. Convolution inverse of A095813.
%F A138520 a(n) = (-1)^n * A228864(n). - _Michael Somos_, Sep 16 2015
%F A138520 a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/5)) / (2 * 5^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 15 2017
%e A138520 G.f. = 1 - q + 2*q^2 - 3*q^3 + 6*q^4 - 11*q^5 + 16*q^6 - 24*q^7 + 38*q^8 + ...
%t A138520 a[ n_] := SeriesCoefficient[ 1 - (EllipticTheta[ 2, 0, q^(5/2)] / EllipticTheta[ 2, 0, q^(1/2)])^2, {q, 0, n}]; (* _Michael Somos_, Sep 16 2015 *)
%t A138520 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^5] / EllipticTheta[ 4, 0, q])^2 QPochhammer[ -q^5, q^5] / QPochhammer[ -q, q]^5, {q, 0, n}]; (* _Michael Somos_, Sep 16 2015 *)
%o A138520 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^10 + A) * ( eta(x^5 + A) / eta(x^2 + A) )^3, n))};
%Y A138520 Cf. A095813, A138519, A138522, A228864.
%K A138520 sign
%O A138520 0,3
%A A138520 _Michael Somos_, Mar 23 2008