A161567 E.g.f. satisfies: A(x) = exp(x*exp(x*A(x)^2)).

1, 1, 3, 22, 233, 3356, 61057, 1343686, 34731377, 1031493880, 34617603041, 1295705404874, 53516386593001, 2417918198462404, 118628419305036929, 6280926119941402486, 356960234149564116833, 21674784895404653181680

Offset: 0

Author

Paul D. Hanna, Jun 14 2009

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 233*x^4/4! +...
log(A(x)) = x*G(x) where G(x) = exp(x*A(x)^2) = e.g.f. of A161565:
G(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 417*x^4/4! + 6201*x^5/5! +...
A(x)^2 = e.g.f. of A161566:
A(x)^2 = 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..365

Crossrefs

Cf. A161565, A161566.

Programs

Mathematica

Flatten[{1,Table[Sum[Binomial[n,k] * (2*(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,binomial(n,k)*(2*(n-k)+1)^(k-1)*k^(n-k))}

PARI

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

Formula

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

E.g.f.: A(x) = F(x)^(1/2) where F(x) = e.g.f. of A161566.

E.g.f.: A(x) = exp(x*G(x)) where G(x) = e.g.f. of A161565.

a(n) ~ n^(n-1) / (2*exp(n)*r^(n+1/2)), 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