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A215648 G.f. satisfies: A(x) = 1 + x*A(x)^2 + 3*x^2*A'(x)*A(x).

%I A215648 #14 Sep 07 2024 16:00:13
%S A215648 1,1,5,44,539,8337,154632,3332640,81711479,2244563555,68272834865,
%T A215648 2278102125040,82749748994500,3250966816344604,137371215935579892,
%U A215648 6213234210869600376,299527133488944917631,15332761842086151881175,830648056455231849827895
%N A215648 G.f. satisfies: A(x) = 1 + x*A(x)^2 + 3*x^2*A'(x)*A(x).
%H A215648 Vaclav Kotesovec, <a href="/A215648/b215648.txt">Table of n, a(n) for n = 0..378</a>
%F A215648 G.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^3]/A(x).
%F A215648 a(n) ~ n! * 3^(n+1) / (Pi*exp(1)). - _Vaclav Kotesovec_, Aug 24 2017
%F A215648 a(0) = 1; a(n) = Sum_{k=0..n-1} (3*k+1) * a(k) * a(n-k-1). - _Ilya Gutkovskiy_, Sep 05 2024
%e A215648 G.f.: A(x) = 1 + x + 5*x^2 + 44*x^3 + 539*x^4 + 8337*x^5 + 154632*x^6 +...
%e A215648 Related expansions:
%e A215648 A(x)^2 = 1 + 2*x + 11*x^2 + 98*x^3 + 1191*x^4 + 18192*x^5 + 333264*x^6 +...
%e A215648 A'(x)*A(x) = 1 + 11*x + 147*x^2 + 2382*x^3 + 45480*x^4 + 999792*x^5 +...
%e A215648 where A(x) = 1 + x*A(x)^2 + 3*x^2*A'(x)*A(x).
%o A215648 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*deriv(x*A^3)/(A+x*O(x^n)));polcoeff(A,n)}
%o A215648 for(n=0,30,print1(a(n),", "))
%Y A215648 Cf. A112936, A112938, A218168.
%K A215648 nonn
%O A215648 0,3
%A A215648 _Paul D. Hanna_, Mar 09 2013