A217616 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(2-x)^(n-k).

1, 2, 4, 12, 38, 116, 360, 1144, 3670, 11836, 38392, 125160, 409628, 1345000, 4428752, 14618608, 48356838, 160260332, 532009688, 1768729736, 5888250996, 19626282328, 65489004464, 218743627408, 731311554044, 2447018357208, 8194289368240, 27459924376592

Offset: 0

Author

Paul D. Hanna, Oct 09 2012

Keywords

• nonn

Comments

Radius of convergence of g.f. is r = 1-sqrt(2)/2 = 0.2928932...

More generally, given

A(x) = Sum_{n>=1} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k),

then A(x) = 1 / sqrt( (1 - t*x + 2*x^2)^2 - 4*x^2 )

and the radius of convergence r satisfies: (1-r)^2 = r*(t-r) for t>0.

Examples

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 38*x^4 + 116*x^5 + 360*x^6 +...
where the g.f. equals the series:
A(x) = 1 +
x*((2-x) + x) +
x^2*((2-x)^2 + 2^2*x*(2-x) + x^2) +
x^3*((2-x)^3 + 3^2*x*(2-x)^2 + 3^2*x^2*(2-x) + x^3) +
x^4*((2-x)^4 + 4^2*x*(2-x)^3 + 6^2*x^2*(2-x)^2 + 4^2*x^3*(2-x) + x^4) +
x^5*((2-x)^5 + 5^2*x*(2-x)^4 + 10^2*x^2*(2-x)^3 + 10^2*x^3*(2-x)^2 + 5^2*x^4*(2-x) + x^5) +...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A217615, A217617, A217461, A216434.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[(1-2*x+2*x^2)^2-4*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 17 2013 *)

PARI

{a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k*(2-x)^(m-k) + x*O(x^n))), n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: A(x) = 1 / sqrt( (1 - 2*x + 2*x^2)^2 - 4*x^2 ).

G.f.: A(x) = 1 / sqrt( (1-4*x+2*x^2)*(1+2*x^2) ).

G.f. satisfies: A(x) = [1 + 2*x^2*Sum_{n>=0} A000108(n)*(-x*A(x))^(2*n)] / (1-2*x+2*x^2) where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan

numbers.

Recurrence: n*a(n) = 2*(2*n-1)*a(n-1) - 4*(n-1)*a(n-2) + 4*(2*n-3)*a(n-3) - 4*(n-2)*a(n-4). - Vaclav Kotesovec, Sep 17 2013

a(n) ~ sqrt(8+6*sqrt(2)) * (2+sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Sep 17 2013