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%I #19 Nov 14 2023 04:12:36
%S 1,1,3,1,2,3,0,1,1,2,0,3,2,0,6,1,2,1,0,2,0,0,0,3,3,2,3,0,2,6,0,1,0,2,
%T 0,1,2,0,6,2,2,0,0,0,2,0,0,3,1,3,6,2,2,3,0,0,0,2,0,6,2,0,0,1,4,0,0,2,
%U 0,0,0,1,2,2,9,0,0,6,0,2,1,2,0,0,4,0,6,0,2,2,0,0,0,0,0,3,2,1,0,3,2,6,0,2,0
%N Expansion of (theta_3(q)^2 + 3 * theta_3(q^3)^2) / 4 - 1 in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.53).
%H G. C. Greubel, <a href="/A163746/b163746.txt">Table of n, a(n) for n = 1..5000</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 psi(q) * psi(q^2) * chi(q^3) * chi(-q^6) - 1 in powers of q where psi(), chi() are Ramanujan theta functions.
%F Expansion of eta(q^2) * eta(q^4)^2 * eta(q^6)^3 / (eta(q) * eta(q^3) * eta(q^12)^2) - 1 in powers of q.
%F Moebius transform is period 12 sequence [ 1, 0, 2, 0, 1, 0, -1, 0, -2, 0, -1, 0, ...].
%F a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2-(-1)^e, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) == (1+(-1)^e)/2 if p == 3 (mod 4). [corrected by _Amiram Eldar_, Nov 14 2023]
%F G.f.: Sum_{k>0} (x^k + x^(3*k)) / (1 - x^(2*k) + x^(4*k)).
%F a(n) = A125061(n) unless n=0. a(12*n + 7) = a(12*n + 11) = 0.
%F a(2*n) = a(n). a(2*n + 1) = A138741(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). - _Michael Somos_, Sep 02 2015
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - _Amiram Eldar_, Nov 14 2023
%e G.f. = q + q^2 + 3*q^3 + q^4 + 2*q^5 + 3*q^6 + q^8 + q^9 + 2*q^10 + 3*q^12 + ...
%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^Quotient[#, 6] {1, 0, 2, 0, 1, 0}[[Mod[#, 6, 1]]] &]]; (* _Michael Somos_, Sep 02 2015 *)
%t a[ n_] := If[ n < 1, 0, Times @@ (Which[# < 3, 1, # == 3, Mod[#2, 2] 2 + 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger @ n)]; (* _Michael Somos_, Sep 02 2015 *)
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^2 + 3 EllipticTheta[ 3, 0, q^3]^2) / 4 - 1, {q, 0, n}]; (* _Michael Somos_, Sep 02 2015 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, ((d%2) * ((d%3==0) + 1)) * (-1)^(d\6)))};
%o (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p==3, e%2*2 + 1, p%4==1, e+1, 1-e%2)))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 * eta(x^6 + A)^3 / (eta(x + A) * eta(x^3 + A) * eta(x^12 + A)^2) - 1, n))};
%Y Cf. A019669, A122856, A122865, A125061, A138741.
%Y Cf. A000122, A000700, A010054, A121373.
%K nonn,easy,mult
%O 1,3
%A _Michael Somos_, Aug 03 2009
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