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A116916 Expansion of q^(-1/8) * (eta(q)^3 + 3 * eta(q^9)^3) in powers of q^3.

%I A116916 #28 Jun 07 2025 05:16:28
%S A116916 1,5,-7,0,0,-11,0,13,0,0,0,0,17,0,0,-19,0,0,0,0,0,0,-23,0,0,0,25,0,0,
%T A116916 0,0,0,0,0,0,29,0,0,0,0,-31,0,0,0,0,0,0,0,0,0,0,-35,0,0,0,0,0,37,0,0,
%U A116916 0,0,0,0,0,0,0,0,0,0,41,0,0,0,0,0,0,-43,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-47,0,0,0,0,0
%N A116916 Expansion of q^(-1/8) * (eta(q)^3 + 3 * eta(q^9)^3) in powers of q^3.
%C A116916 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A116916 Seiichi Manyama, <a href="/A116916/b116916.txt">Table of n, a(n) for n = 0..1000</a>
%H A116916 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A116916 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A116916 Expansion of f(-x) * a(x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.
%F A116916 Expansion of f(-x)^3 + 3 * x * f(-x^9)^3 in powers of x^3 where f() is a Ramanujan theta function.
%F A116916 G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 4608^(1/2) (t / i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A202394.
%F A116916 G.f.: Sum_{k in Z} (-1)^k * (6*k + 1) * x^(k * (3*k + 1) / 2).
%F A116916 a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = 5 * a(n).
%F A116916 a(n) = A010816(3*n).
%e A116916 1 + 5*x - 7*x^2 - 11*x^5 + 13*x^7 + 17*x^12 - 19*x^15 - 23*x^22 + 25*x^26 + ...
%e A116916 q + 5*q^25 - 7*q^49 - 11*q^121 + 13*q^169 + 17*q^289 - 19*q^361 +...
%t A116916 a[0] = 1; a[n_] := SeriesCoefficient[QPochhammer[x + x*O[x]^(3n)]^3 + 3x * QPochhammer[x^9 + O[x]^(3n)]^3, 3n]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Nov 06 2015, adapted from PARI *)
%t A116916 a[ n_] := With[ {m = Sqrt[ 24 n + 1]}, If[ IntegerQ[ m], m KroneckerSymbol[ 3, m] KroneckerSymbol[ -3, m], 0]]; (* _Michael Somos_, Apr 27 2018 *)
%o A116916 (PARI) {a(n) = if( issquare( 24*n + 1, &n), n * kronecker( 3, n) * kronecker( -3, n))};
%o A116916 (PARI) {a(n) = if( n<1, n==0, n*=3; polcoeff( eta(x + x * O(x^n))^3 + 3 * x * eta(x^9 + x * O(x^n))^3, n))};
%Y A116916 Cf. A010816, A202394.
%Y A116916 Cf. A282937, A282941.
%K A116916 sign
%O A116916 0,2
%A A116916 _Michael Somos_, Feb 26 2006