A261252 - OEIS

%I #11 Mar 12 2021 22:24:48

%S 1,-1,2,-4,7,-10,14,-22,33,-45,62,-88,122,-163,216,-290,386,-502,650,

%T -846,1093,-1393,1768,-2248,2844,-3565,4454,-5566,6927,-8566,10562,

%U -13014,15986,-19543,23832,-29032,35272,-42700,51578,-62226,74906,-89909,107712

%N Expansion of f(-x^3) * f(-x^6) / (f(x) * f(-x^4)) in powers of x where f() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A261252/b261252.txt">Table of n, a(n) for n = 0..2500</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 f(-x^3)^3 / (f(-x^2)^2 * f(x, x^2)) in powers of x where f(,) is a Ramanujan theta function.

%F Expansion of (chi(-x^3)^3 / chi(-x)) * (psi(x^3) / psi(x))^2 in powers of x where chi(), psi() are Ramanujan theta functions.

%F Expansion of q^(-1/6) * eta(q) * eta(q^3) * eta(q^6) / eta(q^2)^3 in powers of q.

%F Euler transform of period 6 sequence [ -1, 2, -2, 2, -1, 0, ...].

%F Given g.f. A(x), then B(q) = q * A(q^6) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (3*u^2 - v)^3 * v^3 - 4 * u^2 * (v - u^2) * (2*u^2 - v).

%F a(n) = A261251(3*n).

%F Convolution inverse is A132179.

%F a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (6^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 06 2018

%e G.f. = 1 - x + 2*x^2 - 4*x^3 + 7*x^4 - 10*x^5 + 14*x^6 - 22*x^7 + ...

%e G.f. = q - q^7 + 2*q^13 - 4*q^19 + 7*q^25 - 10*q^31 + 14*q^37 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] QPochhammer[ x^6] / (QPochhammer[ -x] QPochhammer[ x^4]), {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x^2 + A)^3, n))};

%Y Cf. A132179, A261251.

%K sign

%O 0,3

%A _Michael Somos_, Aug 12 2015