A217617 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(3-x)^(n-k).

1, 3, 9, 33, 133, 549, 2295, 9711, 41505, 178749, 774387, 3370995, 14733043, 64608555, 284143257, 1252749777, 5535201733, 24503713893, 108659076723, 482566381299, 2146042722591, 9555487997247, 42594294578949, 190060286569677, 848858809506279, 3794468370955587

Offset: 0

Author

Paul D. Hanna, Oct 09 2012

Keywords

• nonn

Comments

Radius of convergence of g.f. is r = (5-sqrt(17))/4 = 0.21922359...

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 + 3*x + 9*x^2 + 33*x^3 + 133*x^4 + 549*x^5 + 2295*x^6 +...
where the g.f. equals the series:
A(x) = 1 +
x*((3-x) + x) +
x^2*((3-x)^2 + 2^2*x*(3-x) + x^2) +
x^3*((3-x)^3 + 3^2*x*(3-x)^2 + 3^2*x^2*(3-x) + x^3) +
x^4*((3-x)^4 + 4^2*x*(3-x)^3 + 6^2*x^2*(3-x)^2 + 4^2*x^3*(3-x) + x^4) +
x^5*((3-x)^5 + 5^2*x*(3-x)^4 + 10^2*x^2*(3-x)^3 + 10^2*x^3*(3-x)^2 + 5^2*x^4*(3-x) + x^5) +...
		

Crossrefs

Cf. A217615, A217616, A217461, A216454.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[(1-3*x+2*x^2)^2-4*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 16 2013 *)

PARI

{a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k*(3-x)^(m-k) + x*O(x^n))), n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: A(x) = 1 / sqrt( (1 - 3*x + 2*x^2)^2 - 4*x^2 ).

G.f.: A(x) = 1 / sqrt( (1-x+2*x^2)*(1-5*x+2*x^2) ).

G.f. satisfies: A(x) = [1 + 2*x^2*Sum_{n>=0} A000108(n)*(-x*A(x))^(2*n)] / (1-3*x+2*x^2) where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan

numbers.

Recurrence: n*a(n) = 3*(2*n-1)*a(n-1) - 9*(n-1)*a(n-2) + 6*(2*n-3)*a(n-3) - 4*(n-2)*a(n-4). - Vaclav Kotesovec, Sep 16 2013

a(n) ~ 2*((5+sqrt(17))/2)^n/sqrt((42*sqrt(17)-170)*Pi*n). - Vaclav Kotesovec, Sep 16 2013