A195587 a(n) = A163659(n^2), where A163659 is the logarithmic derivative of Stern's diatomic series (A002487).

1, 7, -2, 31, 1, -14, 1, 127, -2, 7, 1, -62, 1, 7, -2, 511, 1, -14, 1, 31, -2, 7, 1, -254, 1, 7, -2, 31, 1, -14, 1, 2047, -2, 7, 1, -62, 1, 7, -2, 127, 1, -14, 1, 31, -2, 7, 1, -1022, 1, 7, -2, 31, 1, -14, 1, 127, -2, 7, 1, -62, 1, 7, -2, 8191, 1, -14, 1, 31, -2, 7, 1, -254, 1, 7, -2, 31, 1, -14, 1, 511, -2, 7, 1, -62, 1, 7, -2, 127, 1, -14, 1, 31, -2, 7, 1, -4094

Offset

1,2

Comments

Multiplicative because A163659 is. - Andrew Howroyd, Jul 26 2018

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Formula

L.g.f.: log(1+x+x^2) + Sum_{n>=0} 3*2^n * log(1 + x^(2*2^n) + x^(4*2^n)) = Sum_{n>=1} a(n)*x^n/n. - Paul D. Hanna, May 04 2014

G.f.: x*(1+2*x)/(1+x+x^2) + Sum_{n>=0} 6*4^n * x^(2*2^n) * (1 + 2*x^(2*2^n)) / (1 + x^(2*2^n) + x^(4*2^n)). - Paul D. Hanna, May 04 2014

Dirichlet g.f.: zeta(s) * (1 - 3^(1-s)) * (2^s + 2) / (2^s - 4). - Amiram Eldar, Oct 24 2023

Example

L.g.f.: L(x) = x + 7*x^2/2 - 2*x^3/3 + 31*x^4/4 + x^5/5 - 14*x^6/6 + x^7/7 + 127*x^8/8 +...
where
exp(L(x)) = 1 + x + 4*x^2 + 3*x^3 + 15*x^4 + 12*x^5 + 37*x^6 + 25*x^7 +...

Mathematica

a[n_] := Times @@ (Function[{p, e}, Which[p == 2, 2^(e+1) - 1, p == 3, -2, True, 1]] @@@ FactorInteger[n^2]);
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019 *)

Prog

(PARI) {A163659(n)=if(n<1, 0, if(n%3, 1, -2)*sigma(2^valuation(n, 2)))}
{a(n)=A163659(n^2)}
for(n=1, 64, print1(a(n), ", "))
(PARI) {a(n)=local(X=x+x*O(x^n), A); A=log(1+X+X^2) + sum(k=0, #binary(n), 3*2^k*log(1 + X^(2*2^k) + X^(4*2^k))); n*polcoeff(A, n)}
for(n=1, 64, print1(a(n), ", "))

Crossrefs

Cf. A195586, A163659, A237649.

Sequence in context: A282609 A282450 A280337 * A096900 A282799 A222555

Adjacent sequences: A195584 A195585 A195586 * A195588 A195589 A195590

Keyword

sign,mult

Author

Paul D. Hanna, Sep 20 2011

Status

approved