A217464 L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k).

1, 1, 4, 5, 6, 16, 29, 45, 94, 186, 331, 644, 1275, 2409, 4644, 9117, 17630, 34162, 66843, 130390, 254153, 497487, 974166, 1906860, 3739131, 7338839, 14406214, 28303525, 55651262, 109465176, 215431773, 424229309, 835747510, 1647145386, 3247768579, 6406290590

Offset: 1

Author

Paul D. Hanna, Oct 03 2012

Keywords

• nonn

Examples

			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 +...
such that the l.g.f. equals the series:
L(x) = ((1-x) + x)*x +
((1-x)^2 + 2^2*x*(1-x) + x^2)*x^2/2 +
((1-x)^3 + 3^2*x*(1-x)^2 + 3^2*x^2*(1-x) + x^3)*x^3/3 +
((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 +
((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 +...
where exponentiation yields the g.f. of A216604:
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 +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A216604.

Programs

Mathematica

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 *)

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)}

PARI

{a(n)=n*polcoeff(log(2/(1-x+sqrt((1-x)^2-4*x^3*(1-x) +x*O(x^n)))), n)}
for(n=1,40,print1(a(n),", "))

Formula

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

a(n) ~ 2^n/sqrt(Pi*n). - Vaclav Kotesovec, Mar 06 2014