A217464 - OEIS

%I #10 Feb 13 2017 14:53:02

%S 1,1,4,5,6,16,29,45,94,186,331,644,1275,2409,4644,9117,17630,34162,

%T 66843,130390,254153,497487,974166,1906860,3739131,7338839,14406214,

%U 28303525,55651262,109465176,215431773,424229309,835747510,1647145386,3247768579,6406290590

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

%H G. C. Greubel, <a href="/A217464/b217464.txt">Table of n, a(n) for n = 1..1000</a>

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

%F a(n) ~ 2^n/sqrt(Pi*n). - _Vaclav Kotesovec_, Mar 06 2014

%e L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 + 5*x^4/4 + 6*x^5/5 + 16*x^6/6 + 29*x^7/7 +...

%e such that the l.g.f. equals the series:

%e L(x) = ((1-x) + x)*x +

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

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

%e ((1-x)^4 + 4^2*x*(1-x)^3 + 6^2*x^2*(1-x)^2 + 4^2*x^3*(1-x) + x^4)*x^4/4 +

%e ((1-x)^5 + 5^2*x*(1-x)^4 + 10^2*x^2*(1-x)^3 + 10^2*x^3*(1-x)^2 + 5^2*x^4*(1-x) + x^5)*x^5/5 +...

%e where exponentiation yields the g.f. of A216604:

%e exp(L(x)) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 12*x^7 + 19*x^8 + 33*x^9 + 59*x^10 + 102*x^11 + 181*x^12 +...+ A216604(n)*x^n +...

%t Rest[CoefficientList[Series[-Log[(1-x+Sqrt[(1-x)^2-4*x^3*(1-x)])/2],{x,0,20}],x]*Range[0,20]] (* _Vaclav Kotesovec_, Mar 06 2014 *)

%o (PARI) {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*sum(k=0, m, binomial(m, k)^2*x^k*(1-x)^(m-k) + x*O(x^n))), n)}

%o (PARI) {a(n)=n*polcoeff(log(2/(1-x+sqrt((1-x)^2-4*x^3*(1-x) +x*O(x^n)))), n)}

%o for(n=1,40,print1(a(n),", "))

%Y Cf. A216604.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Oct 03 2012