A122071 Sum over divisors d of 2n+1 of Kronecker(-18/d).

1, 1, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 1, 2, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 4

Offset: 0

Author

Michael Somos, Aug 20 2006

Keywords

• nonn
• easy

References

• Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 83, Eq. (32.58).

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10000

Crossrefs

A035172(2n+1) = a(n).

Cf. A093825.

Programs

Mathematica

a[n_] := DivisorSum[2*n+1, KroneckerSymbol[-18, #] &]; Array[a, 100, 0] (* Amiram Eldar, Dec 16 2023 *)

PARI

{a(n)=if(n<0, 0, n=2*n+1; sumdiv(n, d, kronecker(-18,d)))}

PARI

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^3+A)*eta(x^8+A)*eta(x^12+A)^3/ eta(x+A)/eta(x^4+A)^2/eta(x^6+A)/eta(x^24+A), n))}

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%8<4, e+1, (1+(-1)^e)/2))))))}

PARI

a(n) = sumdiv(2*n+1, d, kronecker(-18, d)); \\ Michel Marcus, Jul 28 2017

Formula

Expansion of q^(-1/2)*eta(q^2)^2*eta(q^3)*eta(q^8)*eta(q^12)^3/ (eta(q)*eta(q^4)^2*eta(q^6)*eta(q^24)) in powers of q.

Euler transform of period 24 sequence [1, -1, 0, 1, 1, -1, 1, 0, 0, -1, 1, -2, 1, -1, 0, 0, 1, -1, 1, 1, 0, -1, 1, -2, ...].

a(n) = b(2n+1) where b(n) is multiplicative and b(2^e)=0^e, b(3^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).

G.f.: Sum_{k>0} x^k(1-x^(4k-2))(1-x^(6k-3))/(1+x^(12k-6)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi/(3*sqrt(2)) = 0.7404804... (A093825). - Amiram Eldar, Dec 16 2023