A192006 a(n) = Sum_{d|n} Kronecker(-14, d) with a(0) = 2.

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, 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, 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

Offset

0,1

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

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.

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).

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

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

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

Example

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 + ...

Mathematica

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

Prog

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

Crossrefs

Cf. A035176.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A090629 A248623 A086412 * A006928 A087890 A245077

Adjacent sequences: A192003 A192004 A192005 * A192007 A192008 A192009

Keyword

nonn,easy,mult

Author

Michael Somos, Jun 22 2011

Status

approved