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A185183 G.f. A(x) satisfies A(x) = 1+x + x^2*[d/dx A(x)^2].

%I A185183 #21 Jul 16 2025 10:03:49
%S A185183 1,1,2,10,72,672,7640,102072,1564864,27064448,521248320,11064781760,
%T A185183 256702399360,6462978471168,175520877380992,5115062135795584,
%U A185183 159227683153536000,5273353734210310144,185143079148664099840,6869062513111759635456
%N A185183 G.f. A(x) satisfies A(x) = 1+x + x^2*[d/dx A(x)^2].
%H A185183 Vaclav Kotesovec, <a href="/A185183/b185183.txt">Table of n, a(n) for n = 0..400</a>
%F A185183 G.f. A(x) satisfies: A(x) = 1+x + 2*x^2*A(x)*A'(x).
%F A185183 a(n) ~ c * n! * 2^n / sqrt(n), where c = 0.493602095524198015213766719826126125048... - _Vaclav Kotesovec_, Feb 21 2014
%F A185183 a(0) = a(1) = 1; a(n) = (n-1) * Sum_{k=0..n-1} a(k) * a(n-k-1). - _Ilya Gutkovskiy_, Jul 05 2020
%F A185183 a(0) = a(1) = 1; a(n) = 2 * Sum_{k=0..n-1} k * a(k) * a(n-1-k). - _Seiichi Manyama_, Jul 16 2025
%e A185183 G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 72*x^4 + 672*x^5 + 7640*x^6 +...
%e A185183 Related series:
%e A185183 A(x)^2 = 1 + 2*x + 5*x^2 + 24*x^3 + 168*x^4 + 1528*x^5 + 17012*x^6 +...
%e A185183 d/dx A(x)^2 = 2 + 10*x + 72*x^2 + 672*x^3 + 7640*x^4 + 102072*x^5 +...
%o A185183 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x+x^2*deriv(A^2+x*O(x^n))); polcoeff(A, n)}
%o A185183 for(n=0,30,print1(a(n),", "))
%Y A185183 Cf. A143916, A218223, A218224, A376125.
%K A185183 nonn
%O A185183 0,3
%A A185183 _Paul D. Hanna_, Mar 12 2012