A035172 - OEIS

%I #47 Nov 16 2023 08:43:32

%S 1,1,1,1,0,1,0,1,1,0,2,1,0,0,0,1,2,1,2,0,0,2,0,1,1,0,1,0,0,0,0,1,2,2,

%T 0,1,0,2,0,0,2,0,2,2,0,0,0,1,1,1,2,0,0,1,0,0,2,0,2,0,0,0,0,1,0,2,2,2,

%U 0,0,0,1,2,0,1,2,0,0,0,0,1,2,2,0,0,2,0,2,2,0,0,0,0,0,0,1,2,1,2,1,0,2,0,0,0

%N a(n) = Sum_{d|n} Kronecker(-18, d).

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

%C For n nonzero, a(n) is nonzero if and only if n is in A002479.

%D Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 81, Eq. (32.51).

%H Antti Karttunen, <a href="/A035172/b035172.txt">Table of n, a(n) for n = 1..10000</a>

%H Alexander Berkovich and Hamza Yesilyurt, <a href="https://arxiv.org/abs/math/0611300">Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms</a>, arXiv:math/0611300 [math.NT], 2006-2007.

%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 * psi(-q^3) * psi(-q^6) * chi(-q^4) / chi(-q) in powers of q where psi(), chi() are Ramanujan theta functions.

%F From _Michael Somos_, Apr 25 2003: (Start)

%F G.f.: x * Product_{k>0} (1 - x^(3*k)) * (1 - x^(24*k)) * (1 + x^k) / (1 + x^(4*k)).

%F Euler transform of period 24 sequence [ 1, 0, 0, -1, 1, -1, 1, 0, 0, 0, 1, -2, 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 1, -2, ...]. (End)

%F Moebius transform is period 24 sequence [ 1, 0, 0, 0, -1, 0, -1 ,0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, ...]. - _Michael Somos_, Jan 28 2006

%F From _Michael Somos_, Aug 04 2006: (Start)

%F Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1,3 (mod 8), a(p^e) = (1 + (-1)^e)/2 if p == 5,7 (mod 8).

%F G.f.: Sum_{k>0} x^k * (1 - x^(4*k)) * (1 - x^(6* k)) / (1 + x^(12*k)). (End)

%F Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -18.

%F G.f.: 1 + Sum{n = -infinity...infinity} (q^n - q^(5n)) / (1 + q^(12n)) (see Berkovich/Yesilyurt). - _Ralf Stephan_, May 14 2007

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(2)) = 0.7404804... (A093825). - _Amiram Eldar_, Nov 16 2023

%t a[n_] := DivisorSum[n, KroneckerSymbol[-18, #]&]; Array[a, 105] (* _Jean-François Alcover_, Nov 14 2015 *)

%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -18, d)))}

%o (PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -18, p) * X))[n])}

%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^24 + A) / eta(x + A) / eta(x^8 + A), n))}

%Y Cf. A002479, A093825, A122071 (odd bisection).

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

%K nonn,easy,mult

%O 1,11

%A _N. J. A. Sloane_