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%I #17 Dec 28 2023 01:50:57
%S 1,1,2,0,1,0,2,2,2,0,0,0,3,1,2,0,0,0,2,2,2,0,2,0,1,2,2,0,0,0,2,0,4,0,
%T 0,0,2,3,0,0,1,0,4,2,2,0,0,0,2,0,2,0,0,0,2,2,2,0,2,0,1,2,4,0,0,0,0,2,
%U 2,0,0,0,4,1,2,0,2,0,2,2,0,0,0,0,3,0,2,0,0,0,2,2,4,0,0,0,2,4,2,0,0,0,4,0,0
%N Excess of number of divisors of 2n+1 of form 12k+1, 12k+5 over those of form 12k+7, 12k+11.
%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. (32.56).
%H Antti Karttunen, <a href="/A125079/b125079.txt">Table of n, a(n) for n = 0..10000</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 q^(-1/2) * eta(q^3)^3 *eta(q^4) * eta(q^12) / (eta(q) * eta(q^6)^2) in powers of q.
%F Expansion of phi(-q^3) * psi(-q^3) / (chi(-q) * chi(-q^2)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
%F Euler transform of period 12 sequence [ 1, 1, -2, 0, 1, 0, 1, 0, -2, 1, 1, -2, ...].
%F a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138745.
%F a(6*n + 3) = a(6*n + 5) = 0. a(6*n) = A002175(n). a(6*n + 1) = a(2*n) = A008441(n). a(6*n + 2) = 2 * A121444(n). a(n) = A035154(2*n + 1).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - _Amiram Eldar_, Dec 28 2023
%e 1 + x + 2*x^2 + x^4 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^12 + x^13 + 2*x^14 + ...
%e q + q^3 + 2*q^5 + q^9 + 2*q^13 + 2*q^15 + 2*q^17 + 3*q^25 + q^27 + 2*q^29 + ...
%t a[ n_] := If[ n < 0, 0, With[{m = 2 n + 1}, Sum[ KroneckerSymbol[ -36, d], { d, Divisors[ m]}]]]
%o (PARI) {a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( -36, d)))}
%o (PARI) {a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( 6, d) * (-1)^(d\12)))}
%o (PARI) {a(n) = if( n<0, 0, if( n%6==1, n\=3, 1); sumdiv( 2*n + 1, d, kronecker( -4, d)) )}
%o (PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p==3, 1, if( p%4==1, e+1, !(e%2)))))))}
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^6 + A)^2), n))}
%Y Cf. A002175, A008441, A019670, A035154, A121444.
%Y Cf. A000122, A000700, A010054, A121373.
%K nonn,easy
%O 0,3
%A _Michael Somos_, Nov 18 2006
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