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A194471 E.g.f. A(x) satisfies A(x) = exp(x) + x*A(x)^2.

%I A194471 #32 Jun 01 2024 11:31:13
%S A194471 1,2,9,79,1065,19401,445933,12389021,403897553,15120448273,
%T A194471 639345572181,30138682861365,1567316344601593,89137628104427033,
%U A194471 5503952108613407933,366697176991277153341,26220726323043177903009,2002962250253424509250081
%N A194471 E.g.f. A(x) satisfies A(x) = exp(x) + x*A(x)^2.
%C A194471 The radius of convergence r of the e.g.f. satisfies: r = exp(-r)/4 = limit (n+1)*a(n)/a(n+1) = 0.203888354702240... with A(r) = 1/(2*r) = 2.452322501352287...
%H A194471 Seiichi Manyama, <a href="/A194471/b194471.txt">Table of n, a(n) for n = 0..356</a>
%F A194471 E.g.f.: A(x) = (1 - sqrt(1 - 4*x*exp(x))) / (2*x).
%F A194471 a(n) = 1 + n*Sum_{k=0..n-1} C(n-1,k)*a(k)*a(n-1-k) for n>=0.
%F A194471 a(n) ~ sqrt(2)*sqrt(1+LambertW(1/4))*n^(n-1)/(4*exp(n)*LambertW(1/4)^(n+1)). - _Vaclav Kotesovec_, Aug 19 2013
%F A194471 a(n) = n! * Sum_{k=0..n} (k+1)^(n-k-1) * binomial(2*k,k)/(n-k)!. - _Vladimir Kruchinin_, Sep 01 2014
%e A194471 E.g.f.: A(x) = 1 + 2*x + 9*x^2/2! + 79*x^3/3! + 1065*x^4/4! +...
%e A194471 Related expansion:
%e A194471 A(x)^2 = 1 + 4*x + 26*x^2/2! + 266*x^3/3! + 3880*x^4/4! + 74322*x^5/5! +...
%e A194471 Illustrate the recurrence:
%e A194471 a(2) = 1 + 2*(1*1*2 + 1*2*1) = 1 + 2*4 = 9;
%e A194471 a(3) = 1 + 3*(1*1*9 + 2*2*2 + 1*9*1) = 1 + 3*26 = 79;
%e A194471 a(4) = 1 + 4*(1*1*79 + 3*2*9 + 3*9*2 + 1*79*1) = 1 + 4*266 = 1065;
%e A194471 a(5) = 1 + 5*(1*1*1065 + 4*2*79 + 6*9*9 + 4*79*2 + 1*1065*1) = 1 + 5*3880 = 19401.
%t A194471 f[0] = 1; f[n_] := f[n] = 1 + n*Sum[ Binomial[n - 1, k]*f[k]*f[n - 1 - k] , {k, 0, n - 1}]; Array[f, 18, 0] (* _Robert G. Wilson v_, Aug 25 2011 *)
%o A194471 (PARI) {a(n)=n!*polcoeff((1 - sqrt(1 - 4*x*exp(x +O(x^(n+2))))) / (2*x),n)}
%o A194471 (PARI) {a(n)=1+n*sum(k=0,n-1,binomial(n-1,k)*a(k)*a(n-1-k))}
%o A194471 (Maxima) a(n):=n!*sum((k+1)^(n-k-1)*binomial(2*k,k)/(n-k)!,k,0,n); /* _Vladimir Kruchinin_, Sep 01 2014 */
%Y A194471 Cf. A000522, A371318, A373324.
%K A194471 nonn
%O A194471 0,2
%A A194471 _Paul D. Hanna_, Aug 24 2011