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A237649 a(n) = A163659(n^3), where A163659 is the logarithmic derivative of Stern's diatomic series (A002487). 4

%I

%S 1,15,-2,127,1,-30,1,1023,-2,15,1,-254,1,15,-2,8191,1,-30,1,127,-2,15,

%T 1,-2046,1,15,-2,127,1,-30,1,65535,-2,15,1,-254,1,15,-2,1023,1,-30,1,

%U 127,-2,15,1,-16382,1,15,-2,127,1,-30,1,1023,-2,15,1,-254,1,15,-2,524287,1,-30,1,127

%N a(n) = A163659(n^3), where A163659 is the logarithmic derivative of Stern's diatomic series (A002487).

%C Multiplicative because A163659 is. - _Andrew Howroyd_, Jul 27 2018

%H Paul D. Hanna, <a href="/A237649/b237649.txt">Table of n, a(n) for n = 1..1000</a>

%F L.g.f.: log(1+x+x^2) + Sum_{n>=0} 7*4^n * log(1 + x^(2*2^n) + x^(4*2^n)) = Sum_{n>=1} a(n)*x^n/n.

%F G.f.: x*(1+2*x)/(1+x+x^2) + Sum_{n>=0} 14*8^n * x^(2*2^n) * (1 + 2*x^(2*2^n)) / (1 + x^(2*2^n) + x^(4*2^n)).

%e L.g.f.: L(x) = x + 15*x^2/2 - 2*x^3/3 + 127*x^4/4 + x^5/5 - 30*x^6/6 + x^7/7 + 1023*x^8/8 +...+ A163659(n^3)*x^n/n +...

%e where

%e exp(L(x)) = 1 + x + 8*x^2 + 7*x^3 + 63*x^4 + 56*x^5 + 329*x^6 + 273*x^7 + 1736*x^8 +...+ A237646(n)*x^n +...

%o (PARI) {A163659(n)=if(n<1, 0, if(n%3, 1, -2)*sigma(2^valuation(n, 2)))}

%o {a(n)=A163659(n^3)}

%o for(n=1,64,print1(a(n),", "))

%o (PARI) {a(n)=local(X=x+x*O(x^n),A);A=log(1+X+X^2) + sum(k=0,#binary(n),7*4^k*log(1 + X^(2*2^k) + X^(4*2^k)));n*polcoeff(A,n)}

%o for(n=1,64,print1(a(n),", "))

%Y Cf. A237646, A163659.

%K sign,mult

%O 1,2

%A _Paul D. Hanna_, May 03 2014

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Last modified January 22 23:50 EST 2022. Contains 350504 sequences. (Running on oeis4.)