%I #29 Nov 21 2023 08:32:39

%S 1,2,2,2,2,4,2,0,2,2,4,0,2,4,0,4,2,4,2,0,4,0,0,0,2,6,4,2,0,4,4,0,2,0,

%T 4,0,2,4,0,4,4,4,0,0,0,4,0,0,2,2,6,4,4,4,2,0,0,0,4,0,4,4,0,0,2,8,0,0,

%U 4,0,0,0,2,4,4,6,0,0,4,0,4,2,4,0,0,8,0,4,0,4,4,0,0,0,0,0,2,4,2,0,6,4,4,0,4

%N Expansion of (phi(q)^2 + phi(q^3)^2) / 2 in powers of q where phi() is a Ramanujan theta function.

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

%D B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 197, Entry 44.

%H G. C. Greubel, <a href="/A122857/b122857.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

%F Expansion of eta(q^2)^3 * eta(q^3)^2 * eta(q^6) / (eta(q)^2 * eta(q^4)* eta(q^12)) in powers of q.

%F Expansion of 2 * psi(q) * psi(q^2) * psi(q^3) / psi(q^6) - phi(q^3)^2 in powers of q. - _Michael Somos_, Jul 09 2013

%F Euler transform of period 12 sequence [ 2, -1, 0, 0, 2, -4, 2, 0, 0, -1, 2, -2, ...].

%F Moebius transform is period 12 sequence [ 2, 0, 0, 0, 2, 0, -2, 0, 0, 0, -2, 0, ...].

%F a(12*n + 7) = a(12*n + 11) = 0.

%F a(n) = 2 * b(n) where b(n) is multiplicative and b(2^e) = b(3^e) = 1, b(p^e) = e+1 if p == 1, 5 (mod 12), a(p^e) == (1-(-1)^e)/2 if p == 7, 11 (mod 12).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A125061.

%F A035154(n) = a(n) / 2 if n > 0. A008441(n) = a(4*n + 1) / 2. A125079(n) = a(2*n + 1) / 2. A113446(3*n + 1) = A002654(3*n + 1) = a(3*n + 1) / 2.

%F a(n) = (-1)^n * A132003(n). Expansion of (phi(-q^3) / phi(-q)) * phi(-q^2) * phi(-q^6) in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Mar 05 2023

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/3 = 2.0943951... (A019693). - _Amiram Eldar_, Nov 21 2023

%e G.f. = 1 + 2*q + 2*q^2 + 2*q^3 + 2*q^4 + 4*q^5 + 2*q^6 + 2*q^8 + 2*q^9 + 4*q^10 + ...

%t a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -36, #] &]]; (* _Michael Somos_, Jul 09 2013 *)

%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^2 + EllipticTheta[ 3, 0, q^3]^2) / 2, {q, 0, n}]; (* _Michael Somos_, Jul 09 2013 *)

%o (PARI) {a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -36, d)))};

%o (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<5, 1, p%12<6, e+1, !(e%2) )))};

%o (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1],

%o [p, e] = A[k, ]; if( p==2, 1, p==3, 1+e%2*2, p%4==1, e+1, !(e%2) )))};

%Y Cf. A002654, A008441, A019693, A035154, A113446, A125079, A125061, A132003.

%Y Cf. A000122, A000700, A010054, A121373.

%K nonn,easy

%O 0,2

%A _Michael Somos_, Sep 14 2006