A217665 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-3*x)^k.

1, 1, 2, 8, 32, 122, 462, 1758, 6718, 25750, 98956, 381196, 1471678, 5693146, 22064296, 85655812, 333035302, 1296684130, 5055195944, 19731318068, 77098776372, 301561031472, 1180608808044, 4626045139116, 18140934734434, 71191952221114, 279576978531644

Offset: 0

Author

Paul D. Hanna, Oct 10 2012

Keywords

• nonn

Comments

Radius of convergence of g.f. A(x) is |x| < 1/4.

More generally, given

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

A(x) = (1-t*x) / sqrt( (1-(t+1)*x)^2*(1+x^2) + (2*t-3)*x^2 - 2*t*(t-1)*x^3 ).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 32*x^4 + 122*x^5 + 462*x^6 + 1758*x^7 +...
where the g.f. equals the series:
A(x) = 1 +
x*(1 + x/(1-3*x)) +
x^2*(1 + 2^2*x/(1-3*x) + x^2/(1-3*x)^2) +
x^3*(1 + 3^2*x/(1-3*x) + 3^2*x^2/(1-3*x)^2 + x^3/(1-3*x)^3) +
x^4*(1 + 4^2*x/(1-3*x) + 6^2*x^2/(1-3*x)^2 + 4^2*x^3/(1-3*x)^3 + x^4/(1-3*x)^4) +
x^5*(1 + 5^2*x/(1-3*x) + 10^2*x^2/(1-3*x)^2 + 10^2*x^3/(1-3*x)^3 + 5^2*x^4/(1-3*x)^4 + x^5/(1-3*x)^5) +...
		

Links

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

Crossrefs

Cf. A217661, A217664, A217666.

Programs

Mathematica

CoefficientList[Series[(1-3*x)/Sqrt[(1-4*x)*(1 - 4*x + 4*x^2 - 4*x^3)], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 17 2014 *)

PARI

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

Formula

G.f.: (1-3*x) / sqrt(1 - 8*x + 20*x^2 - 20*x^3 + 16*x^4).

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

a(n) ~ 4^n / (sqrt(3*Pi*n)). - Vaclav Kotesovec, Feb 17 2014