A162659 E.g.f. A(x) satisfies A(x) = exp(x*A(x*A(x))).

1, 1, 3, 22, 281, 5396, 142297, 4865806, 207407489, 10710044776, 655655874641, 46789973764634, 3840103504940881, 358443042637767868, 37700333788138306937, 4432826052558222878206, 578707468284010393533953, 83384676375176176768112720, 13190759232920144271864441505

Offset: 0

Author

Paul D. Hanna, Jul 09 2009

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 281*x^4/4! + 5396*x^5/5! +...
A(x*A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 777*x^4/4! + 17581*x^5/5! +...
log(A(x)) = x + 2*x^2/2! + 15*x^3/3! + 196*x^4/4! + 3885*x^5/5! + 105486*x^6/6! +...
		

Crossrefs

Cf. A140049.

Programs

Mathematica

terms = 19; A[] = 0; Do[A[x] = Exp[x*A[x*A[x]]] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Mar 24 2025 *)

PARI

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

PARI

/* Log(A(x)) = x*A(x*A(x)) = Sum_{n>=1} L(n)*x^n/n! where: */
{L(n)=if(n<1,0,sum(k=1,n,binomial(n,k)*(n-k)^(k-1)*a(n-k,k)))}

Formula

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! with a(0,m)=1, then

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

...

Let log(A(x)) = x*A(x*A(x)) = Sum_{n>=1} L(n)*x^n/n!, then

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

...

E.g.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n/n! * [D^(n-1) A(x)^n] where operator D F(x) = d/dx x*F(x). - Paul D. Hanna, Mar 05 2013