A161566 E.g.f. satisfies: A(x) = exp(2*x*exp(x*A(x))).

1, 2, 8, 62, 696, 10362, 193036, 4323846, 113288720, 3401106290, 115150465044, 4341507224958, 180422159478424, 8194551731190762, 403871802897954332, 21468380724070186358, 1224364515329753354784, 74574475891799118725346

Offset: 0

Author

Paul D. Hanna, Jun 14 2009

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + 2*x + 8*x^2/2! + 62*x^3/3! + 696*x^4/4! +...
log(A(x)) = 2*x*C(x) where C(x) = exp(x*A(x)) = e.g.f. of A161565:
C(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 417*x^4/4! + 6201*x^5/5! +...
A(x)^(1/2) = e.g.f. of A161567:
A(x)^(1/2) = 1 + x + 3*x^2/2! + 22*x^3/3! + 233*x^4/4! + 3356*x^5/5! +...
		

Links

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

Crossrefs

Cf. A161565, A161567.

Programs

Mathematica

Flatten[{1,Table[Sum[2^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^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(2*x*exp(x*A+O(x^n))));n!*polcoeff(A,n,x)}

Formula

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

E.g.f.: A(x) = B(x)^2 where B(x) = e.g.f. of A161567.

a(n) ~ sqrt(LambertW(1/(2*r))) * n^(n-1) / (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