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%I #19 Dec 29 2023 03:50:07
%S 1,2,0,4,2,0,0,0,2,6,0,4,4,0,0,0,2,4,0,4,0,0,0,0,4,2,0,8,0,0,0,0,2,8,
%T 0,0,6,0,0,0,0,4,0,4,4,0,0,0,4,2,0,8,0,0,0,0,0,8,0,4,0,0,0,0,2,0,0,4,
%U 4,0,0,0,6,4,0,4,4,0,0,0,0,10,0,4,0,0
%N Expansion of phi(q^4) * phi(q^8) + 2 * q *phi(q^2) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%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 phi(q) * phi(q^8) + 4 * q^3 * psi(q^8) * psi(q^16) in powers of q where phi(), psi() are Ramanujan theta functions.
%F a(n) = 2 * b(n) where b(n) is multiplicative and b(2) = 0, b(2^e) = 1 if e>1, b(p^e) = e+1 if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 if p == 5, 7 (mod 8).
%F G.f.: 1 + 2 * Sum_{k>0 & k !=2 (mod 4)} q^k * (1 + q^(2*k)) / (1 + q^(4*k)).
%F a(4*n + 2) = 0. a(2*n + 1) = 2 * A113411(n). a(4*n) = A033715(n).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3*Pi/(4*sqrt(2)) = 1.6660811... . - _Amiram Eldar_, Dec 29 2023
%e G.f. = 1 + 2*q + 4*q^3 + 2*q^4 + 2*q^8 + 6*q^9 + 4*q^11 + 4*q^12 + 2*q^16 + ...
%t a[ n_] := SeriesCoefficient[ 1 + 2 Sum[ Boole[ 2 != Mod[ k, 4]] q^k (1 + q^(2 k)) / (1 + q^(4 k)), {k, n}], {q, 0, n}];
%o (PARI) {a(n) = if( n<1, n==0, 2 * (n%4 != 2) * sumdiv( n, d, kronecker( -2, d)))};
%o (PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k= 1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, e>1, if( p%8<5, e+1, (1 + (-1)^e) / 2)))))};
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A) * eta(x^16 + A))^3 / (eta(x^4 + A) * eta(x^32 + A))^2 + 2 * x * eta(x^4 + A)^5 * eta(x^16 + A)^2 / (eta(x^2 + A)^2 * eta(x^8 + A)^3), n))};
%o (Magma) A := Basis( ModularForms( Gamma1(32), 1), 87); A[1] + 2*A[2] + 4*A[4] + 2*A[5] + 2*A[9] + 6*A[10] + 4*A[12] + 4*A[13] + 4*A[16]; /* _Michael Somos_, Jun 18 2014 */
%Y Cf. A033715, A113411.
%Y Cf. A000122, A000700, A010054, A121373.
%K nonn,easy
%O 0,2
%A _Michael Somos_, Jun 01 2013
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