cp's OEIS Frontend

A202394 Expansion of f(-x)^3 + 9 * x * f(-x^9)^3 in powers of x where f() is a Ramanujan theta function.

%I A202394 #16 Feb 16 2025 08:33:16
%S A202394 1,6,0,5,0,0,-7,0,0,0,-18,0,0,0,0,-11,0,0,0,0,0,13,0,0,0,0,0,0,30,0,0,
%T A202394 0,0,0,0,0,17,0,0,0,0,0,0,0,0,-19,0,0,0,0,0,0,0,0,0,-42,0,0,0,0,0,0,0,
%U A202394 0,0,0,-23,0,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,54,0,0,0,0
%N A202394 Expansion of f(-x)^3 + 9 * x * f(-x^9)^3 in powers of x where f() is a Ramanujan theta function.
%C A202394 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A202394 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A202394 G. C. Greubel, <a href="/A202394/b202394.txt">Table of n, a(n) for n = 0..1000</a>
%H A202394 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A202394 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A202394 Expansion of f(-x^3) * a(x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.
%F A202394 Expansion of q^(-1/8) * (eta(q)^3 + 9 * eta(q^9)^3) in powers of q.
%F A202394 G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 41472^(1/2) (t / i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A116916.
%F A202394 G.f.: Sum_{k} -(-1)^k * (6*k - 1) * x^(3*k*(3*k - 1)/2) + Sum_{k>0} -(-1)^k * 6 * (2*k - 1) * x^(9*k*(k - 1)/2 + 1).
%F A202394 a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = a(9*n + 4) = a(9*n + 7) = 0. a(3*n) = A116916(n). a(9*n + 1) = 6 * A010816(n). a(25*n + 3) = 5 * a(n).
%F A202394 a(n) nonzero if and only if n is a triangular number.
%e A202394 G.f. = 1 + 6*x + 5*x^3 - 7*x^6 - 18*x^10 - 11*x^15 + 13*x^21 + 30*x^28 + ...
%e A202394 G.f. = q + 6*q^9 + 5*q^25 - 7*q^49 - 18*q^81 - 11*q^121 + 13*q^169 + 30*q^225 + ...
%t A202394 a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3, {x, 0, n}]; (* _Michael Somos_, May 26 2014 *)
%o A202394 (PARI) {a(n) = local(m); if( issquare(8*n + 1, &m), (-1)^(m \ 6) * m * ((m%3 == 0) + 1), 0)};
%o A202394 (PARI) {a(n) = local(A); if( n<0, 0,  A = x * O(x^n); polcoeff( eta(x + A)^3 + 9 * x * eta(x^9 + A)^3, n))};
%Y A202394 Cf. A010816, A116916.
%K A202394 sign
%O A202394 0,2
%A A202394 _Michael Somos_, Dec 18 2011