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A259761 Expansion of (phi(x)^2 + phi(x^9)^2) / 2 in powers of x where phi() is a Ramanujan theta function.

%I A259761 #17 Feb 16 2025 08:33:26
%S A259761 1,2,2,0,2,4,0,0,2,4,4,0,0,4,0,0,2,4,4,0,4,0,0,0,0,6,4,0,0,4,0,0,2,0,
%T A259761 4,0,4,4,0,0,4,4,0,0,0,8,0,0,0,2,6,0,4,4,0,0,0,0,4,0,0,4,0,0,2,8,0,0,
%U A259761 4,0,0,0,4,4,4,0,0,0,0,0,4,4,4,0,0,8,0
%N A259761 Expansion of (phi(x)^2 + phi(x^9)^2) / 2 in powers of x where phi() is a Ramanujan theta function.
%C A259761 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A259761 G. C. Greubel, <a href="/A259761/b259761.txt">Table of n, a(n) for n = 0..1000</a>
%H A259761 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A259761 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A259761 phi(x) = 1 + 2*Sum_{m=1..oo} x^(m^2). - _N. J. A. Sloane_, Jan 30 2017
%F A259761 Expansion of phi(x) * phi(x^9) + 2 * x^2 * chi(x^3)^2 * psi(-x^9)^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
%F A259761 a(n) = 2 * b(n) with a(0) = 1 and b() is multiplicative with b(2^e) = 1, b(3^e) = 1 + (-1)^e if e>0, b(p^e) = e+1 if p == 1, 5 (mod 12), (p^e) = (1 + (-1)^e)/2 if p == 7, 11 (mod 12).
%F A259761 a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.
%F A259761 a(2*n) = a(n). a(9*n) = A004018(n). a(6*n + 4) = 2 * A122856(n).
%F A259761 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5*Pi/9 = 1.745329... (= 100 * A019685). - _Amiram Eldar_, Dec 29 2023
%e A259761 G.f. = 1 + 2*x + 2*x^2 + 2*x^4 + 4*x^5 + 2*x^8 + 4*x^9 + 4*x^10 + 4*x^13 + ...
%t A259761 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]^2 + EllipticTheta[ 3, 0, x^9]^2) / 2, {x, 0, n}];
%t A259761 a[ n_] := If[ n < 1, Boole[n == 0], 2 Times @@ ( Which[ # < 3, 1, # == 3, 1 + (-1)^#2, Mod[#, 12] < 6, #2 + 1, True, (1 + (-1)^#2) / 2 ] & @@@ FactorInteger[n])];
%o A259761 (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 1 + (-1)^e, p%12>6, (1 + (-1)^e) / 2, e+1)))};
%o A259761 (Magma) A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + 2*A[2] + 2*A[3] + 2*A[5] + 4*A[6] + 2*A[9] + 4*A[10] + 4*A[11] + 4*A[14] + 2*A[17] + 4*A[18] + 4*A[19];
%Y A259761 Cf. A004018, A019685, A122856.
%Y A259761 Cf. A000122, A000700, A010054, A121373.
%K A259761 nonn,easy
%O A259761 0,2
%A A259761 _Michael Somos_, Jul 04 2015