A217666 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-4*x)^k.

1, 1, 2, 9, 43, 198, 903, 4121, 18840, 86255, 395397, 1814662, 8337729, 38350063, 176574336, 813785593, 3753980313, 17332179596, 80089232683, 370370470791, 1714045215632, 7938075605697, 36787429315319, 170592514889814, 791557946825363, 3674974608196665

Offset: 0

Author

Paul D. Hanna, Oct 10 2012

Keywords

• nonn

Comments

Radius of convergence of g.f. A(x) is |x| < 0.2116085881629750...

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 + 9*x^3 + 43*x^4 + 198*x^5 + 903*x^6 + 4121*x^7 +...
where the g.f. equals the series:
A(x) = 1 +
x*(1 + x/(1-4*x)) +
x^2*(1 + 2^2*x/(1-4*x) + x^2/(1-4*x)^2) +
x^3*(1 + 3^2*x/(1-4*x) + 3^2*x^2/(1-4*x)^2 + x^3/(1-4*x)^3) +
x^4*(1 + 4^2*x/(1-4*x) + 6^2*x^2/(1-4*x)^2 + 4^2*x^3/(1-4*x)^3 + x^4/(1-4*x)^4) +
x^5*(1 + 5^2*x/(1-4*x) + 10^2*x^2/(1-4*x)^2 + 10^2*x^3/(1-4*x)^3 + 5^2*x^4/(1-4*x)^4 + x^5/(1-4*x)^5) +...
		

Crossrefs

Cf. A217661, A217664, A217665.

Programs

PARI

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

Formula

G.f.: (1-4*x) / sqrt(1 - 10*x + 31*x^2 - 34*x^3 + 25*x^4).