%I #27 Feb 16 2025 08:33:03

%S 1,1,1,3,1,2,3,0,1,1,2,0,3,2,0,6,1,2,1,0,2,0,0,0,3,3,2,3,0,2,6,0,1,0,

%T 2,0,1,2,0,6,2,2,0,0,0,2,0,0,3,1,3,6,2,2,3,0,0,0,2,0,6,2,0,0,1,4,0,0,

%U 2,0,0,0,1,2,2,9,0,0,6,0,2,1,2,0,0,4,0,6,0,2,2,0,0,0,0,0,3,2,1,0,3,2,6,0,2

%N Expansion of psi(q) * psi(q^2) * chi(q^3) * chi(-q^6) in powers of q where psi(), chi() are Ramanujan theta functions.

%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="/A125061/b125061.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="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

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

%F Expansion of (theta_3(q)^2 + 3*theta_3(q^3)^2) / 4 in powers of q.

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

%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).

%F G.f.: 1 + Sum_{k>0} (x^k + x^(3*k)) / (1 - x^(2*k) + x^(4*k)).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A122857.

%F a(12*n + 7) = a(12*n + 11) = 0. a(2*n) = a(n). a(2*n + 1) = A138741(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 1) = A008441(n). a(4*n + 3) = 3 * A008441(n). a(6*n + 1) = A002175(n). a(6*n + 5) = 2 * A121444(n). a(8*n + 1) = A113407(n). a(8*n + 3) = 3 * A113407(n). a(8*n + 5) = 2 * A053692(n). a(8*n + 7) = 6 * A053692(n). a(9*n) = A125061(n).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - _Amiram Eldar_, Nov 24 2023

%e G.f. = 1 + 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 s = (EllipticTheta[3, 0, q]^2 + 3*EllipticTheta[3, 0, q^3]^2)/4 + O[q]^105; CoefficientList[s, q] (* _Jean-François Alcover_, Dec 07 2015, from 2nd formula *)

%o (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, ((d%2) * ((d%3==0)+1)) * (-1)^(d\6)))};

%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) )))};

%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), n))};

%Y Cf. A002175, A008441, A019669, A053692, A113407, A121444, A122856, A122857, A122865, A125061, A138741, A138745, A138746, A138949, A138950, A138951, A138952, A163746.

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

%K nonn,easy,mult,changed

%O 0,4

%A _Michael Somos_, Nov 18 2006