A159599 E.g.f.: A(x) = exp( Sum_{n>=1} [ D^n exp(x) ]^n/n ), where differential operator D = x*d/dx.

1, 1, 4, 27, 304, 5685, 177486, 9305821, 807656872, 113141689065, 25091265489130, 8644033129800321, 4584172093683770820, 3704744323753306881229, 4538175408875808587259022, 8381136688938251234193247485

Offset: 0

Author

Paul D. Hanna, May 05 2009, May 22 2009

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 304*x^4/4! +...
log(A(x)) = x + 3*x^2/2! + 17*x^3/3! + 190*x^4/4! + 3889*x^5/5! +...
log(A(x)) = (D^1 e^x) + (D^2 e^x)^2/2 + (D^3 e^x)^3/3 +...
D^1 exp(x) = (1)*x*exp(x);
D^2 exp(x) = (1 + x)*x*exp(x);
D^3 exp(x) = (1 + 3*x + x^2)*x*exp(x);
D^4 exp(x) = (1 + 7*x + 6*x^2 + x^3)*x*exp(x);
D^5 exp(x) = (1 + 15*x + 25*x^2 + 10*x^3 + x^4)*x*exp(x); ...
D^n exp(x) = n-th iteration of operator D = x*d/dx on exp(x) equals the g.f. of the n-th row of triangle A008277 (S2(n,k)) times x*exp(x), and so is related to the n-th Bell number.
		

Crossrefs

Cf. A159596, A008277 (S2(n, k)), A000110 (Bell).

Programs

PARI

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n, sum(k=1,n,k^m*x^k/k!+x*O(x^n))^m/m))); n!*polcoeff(A,n)}

Formula

E.g.f.: A(x) = exp( Sum_{n>=1} [ Sum_{k>=1} k^n*x^k/k! ]^n/n ).