A217461 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(4-x)^(n-k).

1, 4, 16, 72, 350, 1768, 9120, 47696, 251974, 1341560, 7186912, 38694000, 209187884, 1134838736, 6174666560, 33681995936, 184138474182, 1008642036184, 5534504908640, 30415064058160, 167378028670788, 922251663816368, 5087347689155264, 28091877168106592

Offset: 0

Author

Paul D. Hanna, Oct 10 2012

Keywords

• nonn

Comments

Radius of convergence of g.f. is r = (3-sqrt(7))/2 = 0.1771243444...

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 + 4*x + 16*x^2 + 72*x^3 + 350*x^4 + 1768*x^5 +...
where the g.f. equals the series:
A(x) = 1 +
x*((4-x) + x) +
x^2*((4-x)^2 + 2^2*x*(4-x) + x^2) +
x^3*((4-x)^3 + 3^2*x*(4-x)^2 + 3^2*x^2*(4-x) + x^3) +
x^4*((4-x)^4 + 4^2*x*(4-x)^3 + 6^2*x^2*(4-x)^2 + 4^2*x^3*(4-x) + x^4) +
x^5*((4-x)^5 + 5^2*x*(4-x)^4 + 10^2*x^2*(4-x)^3 + 10^2*x^3*(4-x)^2 + 5^2*x^4*(4-x) + x^5) +...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A217615, A217616, A217617.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[(1-2*x+2*x^2)*(1-6*x+2*x^2)], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

PARI

{a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k*(4-x)^(m-k) + x*O(x^n))), n)}
for(n=0, 40, print1(a(n), ", "))

Formula

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

G.f.: A(x) = 1 / sqrt( (1-2*x+2*x^2)*(1-6*x+2*x^2) ).

G.f. satisfies: A(x) = [1 + 2*x^2*Sum_{n>=0} A000108(n)*(-x*A(x))^(2*n)] / (1-4*x+2*x^2) where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan numbers.

Recurrence: n*a(n) = 4*(2*n-1)*a(n-1) - 16*(n-1)*a(n-2) + 8*(2*n-3)*a(n-3) - 4*(n-2)*a(n-4). - Vaclav Kotesovec, Oct 24 2012

a(n) ~ sqrt(147+56*sqrt(7))*(3+sqrt(7))^n/(14*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 24 2012