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A173610 Bisections are {b(n)*b(n+1), n>=0} (shifted) and {b(n)^2, n>=0} where {b(n), n>=0} is the self-convolution of this sequence, with a(0)=1. 3

%I #2 Mar 30 2012 18:37:21

%S 1,1,2,4,10,25,60,144,384,1024,2752,7396,19436,51076,132888,345744,

%T 938448,2547216,6977712,19114384,52748180,145564225,402295360,

%U 1111822336,3045774336,8343726336,22798000896,62292173056,169191995264

%N Bisections are {b(n)*b(n+1), n>=0} (shifted) and {b(n)^2, n>=0} where {b(n), n>=0} is the self-convolution of this sequence, with a(0)=1.

%C The limit a(n+1)/a(n) seems to exist, with a value near 2.81...

%F G.f. satisfies: (A(x) + A(-x))/2 = 1 + x*Sum_{n>=0} b(n)*b(n+1)*x^(2n+1) and

%F (A(x) - A(-x))/2 = x*Sum_{n>=0} b(n)^2*x^(2n) where Sum_{n>=0} b(n)*x^n = A(x)^2 with a(0)=1.

%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 25*x^5 + 60*x^6 +...

%e The square of the g.f. begins:

%e A(x)^2 = 1 + 2*x + 5*x^2 + 12*x^3 + 32*x^4 + 86*x^5 +...+ b(n)*x^n +...

%e where the odd bisection yields the series with squared coefficients:

%e (A(x) - A(-x))/2 = x + 4*x^3 + 25*x^5 + 144*x^7 + 1024*x^9 +...+ b(n)^2*x^(2n+1) +...

%e and the even bisection yields the series:

%e (A(x) + A(-x))/2 = 1 + 2*x^2 + 10*x^4 + 60*x^6 + 384*x^8 +...+ b(n)*b(n+1)*x^(2n+2) +...

%o (PARI) {a(n)=local(A=1+x,B); for(i=1,n,B=(A+x*O(x^n))^2;A=1+x*sum(m=0,n\2,polcoeff(B,m)*polcoeff(B,m+1)*x^(2*m+1)) +x*sum(m=0,n\2,polcoeff(B,m)^2*x^(2*m)));polcoeff(A,n)}

%Y Cf. A173611 (square), bisections: A173612, A173613.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 22 2010

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