A132965 - OEIS

%I #15 Mar 12 2021 22:24:44

%S 1,2,1,2,4,4,5,6,8,10,12,14,17,22,24,30,36,40,48,56,65,76,88,100,116,

%T 134,152,174,200,226,257,292,328,372,420,472,532,598,668,750,840,936,

%U 1045,1166,1296,1442,1604,1776,1972,2186,2416,2672,2952,3256,3592,3960

%N Expansion of f(-q^8) * chi(q)^2 in powers of q where f(), 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="/A132965/b132965.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 q^(-1/4) * eta(q^2)^4 * eta(q^8) / (eta(q)^2 * eta(q^4)^2) in powers of q.

%F Euler transform of period 8 sequence [ 2, -2, 2, 0, 2, -2, 2, -1, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 8^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132966.

%F G.f.: Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k)) * (1 + x^(4*k)) / (1 + x^(2*k)).

%F a(n) ~ exp(sqrt(n)*Pi/2) / (4*sqrt(n)). - _Vaclav Kotesovec_, Sep 08 2015

%e G.f. = 1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 8*x^8 + 10*x^9 + ...

%e G.f. = q + 2*q^5 + q^9 + 2*q^13 + 4*q^17 + 4*q^21 + 5*q^25 + 6*q^29 + 8*q^33 + ...

%t nmax = 60; CoefficientList[Series[Product[(1 + x^k)^2 * (1 - x^(2*k)) * (1 + x^(4*k)) / (1 + x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^8] QPochhammer[ -x, x^2]^2, {x, 0, n}]; (* _Michael Somos_, Nov 01 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^8 + A) / (eta(x + A)^2 * eta(x^4 + A)^2), n))};

%Y Cf. A132966.

%K nonn

%O 0,2

%A _Michael Somos_, Aug 23 2007