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%I #29 Oct 24 2023 04:33:39
%S 1,3,-2,7,1,-6,1,15,-2,3,1,-14,1,3,-2,31,1,-6,1,7,-2,3,1,-30,1,3,-2,7,
%T 1,-6,1,63,-2,3,1,-14,1,3,-2,15,1,-6,1,7,-2,3,1,-62,1,3,-2,7,1,-6,1,
%U 15,-2,3,1,-14,1,3,-2,127,1,-6,1,7,-2,3,1,-30,1,3,-2,7,1,-6,1,31,-2,3,1,-14,1
%N L.g.f.: Sum_{n>=1} a(n)*x^n/n = log(S(x)/x) where S(x) is the g.f. of Stern's diatomic series (A002487).
%H G. C. Greubel, <a href="/A163659/b163659.txt">Table of n, a(n) for n = 1..1000</a>
%F a(2^n) = 2^(n+1) - 1 for n>=0.
%F a(n) is multiplicative with a(2^e) = 2^(e+1) - 1, a(3^e) = -2 if e>0, a(p^e) = 1 if p>3. - _Michael Somos_, Feb 14 2011
%F L.g.f.: Sum_{n>=0} log(1 + x^(2^n) + x^(2*2^n)) = Sum_{n>=1} a(n)*x^n/n. - _Paul D. Hanna_, May 04 2014
%F G.f.: Sum_{n>=0} 2^n * x^(2^n) * (1 + 2*x^(2^n)) / (1 + x^(2^n) + x^(2*2^n)). - _Paul D. Hanna_, May 04 2014
%F Dirichlet g.f.: zeta(s) * (1 - 3^(1-s)) / (1 - 2^(1-s)). - _Amiram Eldar_, Oct 24 2023
%e L.g.f.: log(S(x)/x) = x + 3*x^2/2 - 2*x^3/3 + 7*x^4/4 + x^5/5 - 6*x^6/6 +...
%e where S(x) is the g.f. of Stern's diatomic series (A002487):
%e S(x) = x + x^2 + 2*x^3 + x^4 + 3*x^5 + 2*x^6 + 3*x^7 + x^8 + 4*x^9 +...
%t a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 2, 1, # == 2, 2^(#2 + 1) - 1, # == 3, -2, True, 1] & @@@ FactorInteger[n])]; (* _Michael Somos_, Jun 27 2017 *)
%o (PARI) A002487(n)=local(c=1, b=0); while(n>0, if(bitand(n, 1), b+=c, c+=b); n>>=1); b
%o {a(n)=n*polcoeff(log(sum(k=0,n,A002487(k+1)*x^k)+x*O(x^n)),n)}
%o for(n=1, 64, print1(a(n), ", "))
%o (PARI) {a(n) = if( n<1, 0, if( n%3, 1, -2) * sigma(2 ^ valuation(n, 2)))}; /* _Michael Somos_, Feb 14 2011 */
%o (PARI) a(n)=local(X=x+x*O(x^n), A); A=log(1+X+X^2) + sum(k=0, #binary(n), log(1 + X^(2*2^k) + X^(4*2^k))); n*polcoeff(A, n)
%o for(n=1, 64, print1(a(n), ", ")) \\ _Paul D. Hanna_, May 04 2014
%Y Cf. A163658, A195587, A002487.
%K sign,easy,mult
%O 1,2
%A _Paul D. Hanna_, Aug 02 2009
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