A192006 - OEIS

%I #28 Dec 16 2023 09:01:10

%S 2,1,1,2,1,2,2,1,1,3,2,0,2,2,1,4,1,0,3,2,2,2,0,2,2,3,2,4,1,0,4,0,1,0,

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

%U 0,4,2,2,3,0,0,6,2,0,4,2,2,5,0,2,2,0,0,0,0,0,6,2,2,0,0,4,2,0,1,0,3,2,0,0,2

%N a(n) = Sum_{d|n} Kronecker(-14, d) with a(0) = 2.

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

%H G. C. Greubel, <a href="/A192006/b192006.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 * f(q^2) * f(q^7) / (chi(q) * chi(q^14)) + 2 * psi(q^4) * phi(-q^7) * chi(-q^14) / chi(-q^2) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (56 t)) = 56^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

%F a(n) is multiplicative with a(0) = 2, a(p^e) = (1 - q^e) / (1 - q) where q = p * Kronecker( -14, p).

%F a(n) = A035176(n) unless n=0.

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(14) = 1.679251... . - _Amiram Eldar_, Dec 16 2023

%e G.f. = 2 + q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 + q^7 + q^8 + 3*q^9 + 2*q^10 + ...

%t a[ n_] := If[ n < 1, 2 * Boole[ n==0], Sum[ KroneckerSymbol[ -14, d], { d, Divisors[ n]}]];

%o (PARI) {a(n) = if( n<1, 2 * (n==0), sumdiv( n, d, kronecker( -14, d)))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 2 * (eta(x^7 + A) * eta(x^8 + A))^2 / (eta(x^2 + A) * eta(x^28 + A)) + x * eta(x + A) * eta(x^4 + A)^4 * eta(x^14 + A)^4 * eta(x^56 + A) / (eta(x^2 + A)^3* eta(x^7 + A) * eta(x^8 + A) * eta(x^28 + A)^3), n))};

%Y Cf. A035176.

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

%K nonn,easy,mult

%O 0,1

%A _Michael Somos_, Jun 22 2011