A245424 - OEIS

%I #12 Mar 12 2021 22:24:47

%S 1,1,2,2,1,-1,-2,-3,-2,1,4,6,5,1,-5,-11,-12,-7,3,15,22,19,5,-15,-32,

%T -36,-22,8,40,58,50,12,-41,-84,-93,-54,22,103,148,124,32,-96,-200,

%U -219,-128,46,231,330,275,67,-216,-439,-477,-275,107,501,708,590,146

%N Expansion of q^(-1) * f(-q^4, -q^5)^2 / (f(-q, -q^8) * f(-q^2, -q^7)) in powers of q where f() is Ramanujan's two-variable theta function.

%C A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_1(9).

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

%H G. C. Greubel, <a href="/A245424/b245424.txt">Table of n, a(n) for n = -1..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) * f(-q^4, -q^5)^3 * f(-q^3) / (f(-q) * f(-q^9)^3) in powers of q where f() is a Ramanujan theta function.

%F Euler transform of period 9 sequence [ 1, 1, 0, -2, -2, 0, 1, 1, 0, ...].

%F Given g.f. A(q), then 0 = f(A(q), A(q^2)) where f(u, v) = (u^2 - v) * (u*v^2 + v^2 + 1)  - 2 * (u + v + u*v)^2.

%F Given g.f. A(q), then 0 = f(A(q), A(q^3)) where f(u, v) = (v^2 + v + 1) * (u^3 - v)  - 3*u*v * (u + 1) * (v + 2).

%F a(n) = A245421(n) unless n=0.

%e G.f. = 1/q + 1 + 2*q + 2*q^2 + q^3 - q^4 - 2*q^5 - 3*q^6 - 2*q^7 + q^8 + ...

%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^4, q^9] QPochhammer[ q^5, q^9])^3 QPochhammer[ q^3] / (QPochhammer[ q] ), {q, 0, n}];

%t a[ n_] := If[ n < -1, 0, With[{m = n + 1}, SeriesCoefficient[ 1/q Product[ (1 - q^k)^{-1, -1, 0, 2, 2, 0, -1, -1, 0}[[Mod[k, 9, 1]]], {k, m}], {q, 0, n}]]];

%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^4, q^9] QPochhammer[ q^5, q^9])^2 / (QPochhammer[ q^1, q^9] QPochhammer[ q^2, q^9] QPochhammer[ q^7, q^9] QPochhammer[ q^8, q^9] ), {q, 0, n}];

%o (PARI) {a(n) = if( n<-1, 0, n++; polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[0, -1, -1, 0, 2, 2, 0, -1, -1][k%9 + 1]), n))};

%Y Cf. A245421.

%K sign

%O -1,3

%A _Michael Somos_, Jul 21 2014