A224922 O.g.f. satisfies: A(x) = Sum_{n>=0} A000110(n)*x^n*A(x)^n, where A000110 are the Bell numbers.

1, 1, 3, 12, 56, 288, 1586, 9201, 55675, 349159, 2260209, 15063260, 103201968, 726380164, 5252083746, 39029442336, 298340012448, 2348365289852, 19058365017840, 159659978176454, 1382148854813222, 12373696834781918, 114606230432935860, 1098199966940781258

Offset: 0

Author

Paul D. Hanna, Apr 19 2013

Keywords

• nonn

Examples

			O.g.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 56*x^4 + 288*x^5 + 1586*x^6 +...
The o.g.f. satisfies:
(1) A(x) = 1 + x*A(x) + 2*x^2*A(x)^2 + 5*x^3*A(x)^3 + 15*x^4*A(x)^4 + 52*x^5*A(x)^5 + 203*x^6*A(x)^6 +...+ A000110(n)*x^n*A(x)^n +...
(2) A(x) = 1 + x*A(x)/(1-x*A(x)) + x^2*A(x)^2/((1-x*A(x))*(1-2*x*A(x))) + x^3*A(x)^3/((1-x*A(x))*(1-2*x*A(x))*(1-3*x*A(x))) + x^4*A(x)^4/((1-x*A(x))*(1-2*x*A(x))*(1-3*x*A(x))*(1-4*x*A(x))) +...
		

Crossrefs

Cf. A099947, A000110.

Programs

PARI

{a(n)=if(n<0, 0, polcoeff( 1/x*serreverse(x/serlaplace(exp(exp(x+x*O(x^n))-1))), n))}
for(n=0,30,print1(a(n),", "))

PARI

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*A^m/prod(k=1, m,1-k*x*A +x*O(x^n)) )); polcoeff(A, n)}
for(n=0,30,print1(a(n), ", "))

Formula

O.g.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^n / Product_{k=1..n} (1 - k*x*A(x)).

O.g.f. satisfies: A(x) = 1/(1 - x*A(x)/(1 - 1*x*A(x)/(1 - x*A(x)/(1 - 2*x*A(x)/(1 - x*A(x)/(1 - 3*x*A(x)/(1 - x*A(x)/(1 - 4*x*A(x)/(1 - ... ))))))))), a continued fraction.

a(n) = [x^n] ( Sum_{k>=0} A000110(k)*x^k )^(n+1) / (n+1).