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A260676 Expansion of phi(x) * psi(x^30) in powers of x where phi(), chi() are Ramanujan theta functions.

%I A260676 #22 Feb 16 2025 08:33:26
%S A260676 1,2,0,0,2,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,1,2,0,0,
%T A260676 2,0,2,0,0,2,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,2,0,
%U A260676 0,0,0,0,0,0,0,0,0,0,0,2,0,2,0,0,0,0,0
%N A260676 Expansion of phi(x) * psi(x^30) in powers of x where phi(), chi() are Ramanujan theta functions.
%C A260676 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260676 G. C. Greubel, <a href="/A260676/b260676.txt">Table of n, a(n) for n = 0..5000</a>
%H A260676 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260676 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260676 Expansion of q^(-15/4) * eta(q^2)^5 * eta(q^60)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^30)) in powers of q.
%F A260676 Euler transform of a period 60 sequence.
%F A260676 2 * a(n) = A260671(4*n + 15).
%e A260676 G.f. = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + x^30 + 2*x^31 + 2*x^34 + ...
%e A260676 G.f. = q^15 + 2*q^19 + 2*q^31 + 2*q^51 + 2*q^79 + 2*q^115 + q^135 + 2*q^139 + ...
%t A260676 a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, x] QPochhammer[x^60]^2 / QPochhammer[x^30], {x, 0, n}];
%o A260676 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^60 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^30 + A)), n))};
%o A260676 (Magma)
%o A260676 m:=130;
%o A260676 f:= func< q | (&*[( (1-q^(2*n))^5*(1-q^(60*n))^2 )/( (1-q^n)^2*(1-q^(4*n))^2*(1-q^(30*n)) ): n in [1..m+1]]) >;
%o A260676 R<x>:=PowerSeriesRing(Integers(), m);
%o A260676 Coefficients(R!( f(x) )); // _G. C. Greubel_, Feb 02 2023
%o A260676 (SageMath)
%o A260676 m = 130
%o A260676 def f(q): return product( ((1-q^(2*n))^5*(1-q^(60*n))^2)/((1-q^n)^2*(1-q^(4*n))^2*(1-q^(30*n))) for n in range(1,m+2))
%o A260676 def A260676_list(prec):
%o A260676     P.<x> = PowerSeriesRing(ZZ, prec)
%o A260676     return P( f(x) ).list()
%o A260676 A260676_list(m) # _G. C. Greubel_, Feb 02 2023
%Y A260676 Cf. A260671.
%K A260676 nonn
%O A260676 0,2
%A A260676 _Michael Somos_, Nov 14 2015