A161565 E.g.f. satisfies: A(x) = exp(x*exp(2*x*A(x))).

1, 1, 5, 37, 417, 6201, 115393, 2583141, 67643201, 2029868785, 68699859201, 2589393498429, 107580709769569, 4885086832499433, 240716442970201409, 12793428673619226901, 729511897042502788737, 44427614415877495608801

Offset: 0

Author

Paul D. Hanna, Jun 14 2009

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 417*x^4/4! +...
log(A(x)) = x*B(x) where B(x) = exp(2*x*A(x)) = e.g.f. of A161566:
B(x) = 1 + 2*x + 8*x^2/2! + 62*x^3/3! + 696*x^4/4! + 10362*x^5/5! +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..367

Crossrefs

Cf. A161566.

Programs

Mathematica

Flatten[{1,Table[Sum[2^(n-k) * Binomial[n,k] * (n-k+1)^(k-1) * k^(n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 28 2014 *)

PARI

{a(n)=sum(k=0,n,2^(n-k)*binomial(n,k) * (n-k+1)^(k-1) * k^(n-k))}

PARI

{a(n)=local(A=1+x);for(i=0,n,A=exp(x*exp(2*x*A+O(x^n))));n!*polcoeff(A,n,x)}

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * C(n,k) * (n-k+1)^(k-1) * k^(n-k).

a(n) ~ sqrt(LambertW(1/r)) * n^(n-1) / (2*exp(n)*r^(n+1)), where r = 0.256263163133653382... is the root of the equation 1/LambertW(1/r) = -log(2*r^2) - LambertW(1/r). - Vaclav Kotesovec, Feb 28 2014