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.

1, 1, 2, 4, 10, 25, 60, 144, 384, 1024, 2752, 7396, 19436, 51076, 132888, 345744, 938448, 2547216, 6977712, 19114384, 52748180, 145564225, 402295360, 1111822336, 3045774336, 8343726336, 22798000896, 62292173056, 169191995264

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..28.

Formula

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

(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.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 25*x^5 + 60*x^6 +...
The square of the g.f. begins:
A(x)^2 = 1 + 2*x + 5*x^2 + 12*x^3 + 32*x^4 + 86*x^5 +...+ b(n)*x^n +...
where the odd bisection yields the series with squared coefficients:
(A(x) - A(-x))/2 = x + 4*x^3 + 25*x^5 + 144*x^7 + 1024*x^9 +...+ b(n)^2*x^(2n+1) +...
and the even bisection yields the series:
(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) +...

Prog

(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)}

Crossrefs

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

Sequence in context: A123420 A000458 A089928 * A036887 A307578 A151536

Adjacent sequences: A173607 A173608 A173609 * A173611 A173612 A173613

Keyword

nonn

Author

Paul D. Hanna, Feb 22 2010

Status

approved