A217665 - OEIS

%I #8 Feb 19 2014 08:03:05

%S 1,1,2,8,32,122,462,1758,6718,25750,98956,381196,1471678,5693146,

%T 22064296,85655812,333035302,1296684130,5055195944,19731318068,

%U 77098776372,301561031472,1180608808044,4626045139116,18140934734434,71191952221114,279576978531644

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

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

%C More generally, given

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

%C 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 ).

%H Vincenzo Librandi, <a href="/A217665/b217665.txt">Table of n, a(n) for n = 0..200</a>

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

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

%F a(n) ~ 4^n / (sqrt(3*Pi*n)). - _Vaclav Kotesovec_, Feb 17 2014

%e G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 32*x^4 + 122*x^5 + 462*x^6 + 1758*x^7 +...

%e where the g.f. equals the series:

%e A(x) = 1 +

%e x*(1 + x/(1-3*x)) +

%e x^2*(1 + 2^2*x/(1-3*x) + x^2/(1-3*x)^2) +

%e x^3*(1 + 3^2*x/(1-3*x) + 3^2*x^2/(1-3*x)^2 + x^3/(1-3*x)^3) +

%e 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) +

%e 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) +...

%t 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 *)

%o (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)}

%o for(n=0,40,print1(a(n),", "))

%Y Cf. A217661, A217664, A217666.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 10 2012