A260676 - OEIS

%I #21 Feb 02 2023 10:45:40

%S 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 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 0,0,0,0,0,0,0,0,0,0,0,2,0,2,0,0,0,0,0

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

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

%H G. C. Greubel, <a href="/A260676/b260676.txt">Table of n, a(n) for n = 0..5000</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^(-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 Euler transform of a period 60 sequence.

%F 2 * a(n) = A260671(4*n + 15).

%e 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 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 a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, x] QPochhammer[x^60]^2 / QPochhammer[x^30], {x, 0, n}];

%o (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 (Magma)

%o m:=130;

%o 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 R<x>:=PowerSeriesRing(Integers(), m);

%o Coefficients(R!( f(x) )); // _G. C. Greubel_, Feb 02 2023

%o (SageMath)

%o m = 130

%o 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 def A260676_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( f(x) ).list()

%o A260676_list(m) # _G. C. Greubel_, Feb 02 2023

%Y Cf. A260671.

%K nonn

%O 0,2

%A _Michael Somos_, Nov 14 2015