A138517 - OEIS

%I #27 Mar 12 2021 23:54:38

%S 1,4,12,32,76,164,336,656,1228,2228,3932,6768,11408,18872,30688,49152,

%T 77644,121096,186684,284720,429916,643168,953904,1403312,2048784,

%U 2969764,4275656,6116480,8696864,12294680,17285776,24176288,33645132

%N Expansion of (phi(-q^5) / phi(-q))^2 in powers of q where phi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

%H G. C. Greubel, <a href="/A138517/b138517.txt">Table of n, a(n) for n = 0..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^5) / eta(q))^2 * eta(q^2) / eta(q^10) )^2 in powers of q.

%F Euler transform of period 10 sequence [ 4, 2, 4, 2, 0, 2, 4, 2, 4, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v^2) * (u - 1) - 4 * u * v * (v - 1).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (1 - u) * (1 - 5*u) * v * (1 - v) * (1 - 5*v).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138516.

%F G.f.: (Product_{k>0} P(5, x^k) / P(10, x^k))^2 where P(n, x) is the n-th cyclotomic polynomial.

%F a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(3/4) * 5^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 03 2018

%F Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 3/10 + (1/10)*sqrt(5) + (1/10)*sqrt(10 + 6*sqrt(5)). - _Simon Plouffe_, Mar 04 2021

%e 1 + 4*q + 12*q^2 + 32*q^3 + 76*q^4 + 164*q^5 + 336*q^6 + 656*q^7 + ...

%t eta[x_] := x^(1/24)*QPochhammer[x]; A138517[n_] := SeriesCoefficient[ ((eta[q^5]/eta[q])^2*eta[q^2]/eta[q^10])^2, {q, 0, n}]; Table[ A138517[n], {n, 0, 50}] (* _G. C. Greubel_, Sep 29 2017 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^5 + A) / eta(x + A))^2 * eta(x^2 + A) / eta(x^10 + A))^2, n))}

%Y Cf. 4 * A095846(n) = a(n) unless n=0. Convolution inverse of A138518. Convolution square of A138526.

%K nonn

%O 0,2

%A _Michael Somos_, Mar 23 2008