A166897 - OEIS

%I #6 Nov 27 2017 13:19:38

%S 1,3,13,39,126,477,1765,6495,24709,95128,367368,1431453,5620343,

%T 22170543,87858813,349708431,1397003136,5598513261,22502171771,

%U 90681323364,366299212873,1482827487650,6014529069540,24439715146941

%N a(n) = Sum_{k=0..[n/2]} C(n-k,k)^3*n/(n-k), n>=1.

%F Logarithmic derivative of A166896.

%F a(n) ~ sqrt(15) * phi^(3*n + 2) / (6*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Nov 27 2017

%e L.g.f.: L(x) = x + 3*x^2/2 + 13*x^3/3 + 39*x^4/4 + 126*x^5/5 + 477*x^6/6 +...

%e exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 142*x^6 + 459*x^7 +...+ A166896(n)*x^n/n +...

%t Table[Sum[Binomial[n-k,k]^3 n/(n-k),{k,0,Floor[n/2]}],{n,30}] (* _Harvey P. Dale_, Mar 05 2013 *)

%o (PARI) a(n)=sum(k=0,n\2,binomial(n-k,k)^3*n/(n-k))

%Y Cf. A166897, variants: A167539, A166895, A166899.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Nov 23 2009