A168600 E.g.f. A(x) satisfies A(x) = exp( x*A(2*x)^2 ).

1, 1, 9, 265, 19889, 3506801, 1417530745, 1302573091513, 2700478102745057, 12518436654808255585, 128568477648089286062441, 2900655737241126221237790185, 142677722979145454671155940121233, 15200178301599487957128451391538504145

Offset: 0

Author

Paul D. Hanna, Dec 05 2009

Keywords

• nonn

Examples

			E.g.f: A(x) = 1 + x + 9*x^2/2! + 265*x^3/3! + 19889*x^4/4! +...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..76

Crossrefs

Cf. A096538, A168601.

Cf. A360987.

Programs

Maple

F:= A -> A(x) - exp(x*A(2*x)^2):
Extend:= proc(ff)
  local f1x, m, f2, S, R, i;
  f1x:= ff(x); m:= degree(f1x,x);
  f2:= unapply(ff(x) + add(a[i]*x^i,i=m+1..2*m+1),x);
  S:= series(F(f2),x,2*m+2);
  R:= solve(identity(convert(S,polynom),x),{seq(a[i],i=m+1..2*m+1)});
  unapply(subs(R, f2(x)),x);
end proc:
g:= 1:
for iter from 1 to 5 do g:= Extend(g) od:
seq(coeff(g(x),x,j)*j!,j=0..31); # Robert Israel, Feb 22 2019

Mathematica

nmax = 13; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - Exp[x A[2 x]^2] + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. HoldPattern[a[n_] -> k_] :> Set[a[n], k n!];
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)

PARI

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

Formula

a(n) ~ c * n! * 2^(n*(n+1)/2), where c = 0.986274628764911276343959993... - Vaclav Kotesovec, Jul 02 2025

a(0) = 1; a(n) = 2^(n-1) * (n-1)! * Sum_{i, j, k>=0 and i+j+k=n-1} (n-i)/2^i * a(i) * a(j) * a(k)/(i! * j! * k!). - Seiichi Manyama, Jul 06 2025