%I #38 Oct 29 2019 15:54:04

%S 1,1,4,41,871,36137,2885457,443469511,131707909982,75945551138638,

%T 85425571722359386,188277619627892581987,816318863956958720950775,

%U 6986374103851011507327849798,118360360643974268213872443877649,3978536338453184605328853807076468581

%N O.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^n * x^n * A((n+1)*x)^n/n! * exp(-(n+1)*x*A((n+1)*x)).

%C Compare to the LambertW identity:

%C Sum_{n>=0} (n+1)^n * x^n * G(x)^n/n! * exp(-(n+1)*x*G(x)) = 1/(1 - x*G(x)).

%e O.g.f.: A(x) = 1 + x + 4*x^2 + 41*x^3 + 871*x^4 + 36137*x^5 + 2885457*x^6 +...

%e where

%e A(x) = exp(-x*A(x)) + 2*x*A(2*x)*exp(-2*x*A(2*x)) + 3^2*x^2*A(3*x)^2/2!*exp(-3*x*A(3*x)) + 4^3*x^3*A(4*x)^3/3!*exp(-4*x*A(4*x)) + 5^4*x^4*A(5*x)^4/4!*exp(-5*x*A(5*x)) + 6^5*x^5*A(6*x)^5/5!*exp(-6*x*A(6*x)) +...

%e simplifies to a power series in x with integer coefficients.

%t A[_] = 0; m = 16;

%t Do[A[x_] = Exp[-x A[x]] + Sum[(n+1)^n x^n A[(n+1)x]^n/n! Exp[-(n+1) x A[(n+1)x]], {n, 1, m}] + O[x]^m // Normal, {m}];

%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Oct 29 2019 *)

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (k+1)^k*x^k*subst(A, x, (k+1)*x)^k/k!*exp(-(k+1)*x*subst(A, x, (k+1)*x)+x*O(x^n)))); polcoeff(A, n)}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A218672, A209276, A209277.

%Y Cf. A221409, A221410, A221411, A221412, A221413.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 09 2013