cp's OEIS Frontend

A218222 G.f. A(x) satisfies: A(x) = x + x*[d/dx A(x)^2].

%I A218222 #18 Dec 18 2020 11:49:55
%S A218222 1,2,12,112,1360,19872,335104,6359040,133560576,3069007360,
%T A218222 76493880320,2054400577536,59136549994496,1816392567062528,
%U A218222 59305340822814720,2051451257317490688,74958908119819812864,2885480280276224311296,116731741304854533111808
%N A218222 G.f. A(x) satisfies: A(x) = x + x*[d/dx A(x)^2].
%H A218222 Vaclav Kotesovec, <a href="/A218222/b218222.txt">Table of n, a(n) for n = 1..300</a>
%F A218222 a(n) = 2^(n-1)*A088716(n-1) for n>=1, where g.f. F(x) of A088716 satisfies: F(x) = 1 + x*F(x)*d/dx[x*F(x)].
%F A218222 a(n) = n*A112915(n-1) for n>=1.
%F A218222 G.f.: x*d/dx x*G(x), where g.f. G(x) of A112915 satisfies: G(x) = 1 + x*(d/dx[x*G(x)])^2.
%F A218222 a(n) ~ c * n * 2^(n-1) * n!, where c = A238223 = 0.21795078944715106549... - _Vaclav Kotesovec_, Aug 24 2017
%e A218222 G.f.: A(x) = x + 2*x^2 + 12*x^3 + 112*x^4 + 1360*x^5 + 19872*x^6 +...
%e A218222 Related series:
%e A218222 A(x)^2 = x^2 + 4*x^3 + 28*x^4 + 272*x^5 + 3312*x^6 + 47872*x^7 + 794880*x^8 + 14840064*x^9 +...+ A112915(n-1)*x^n +...
%e A218222 d/dx A(x)^2 = 2*x + 12*x^2 + 112*x^3 + 1360*x^4 + 19872*x^5 +...
%p A218222 a:= proc(n) option remember; `if`(n<2, 1,
%p A218222       n*add(a(i)*a(n-i), i=1..n-1))
%p A218222     end:
%p A218222 seq(a(n), n=1..20);  # _Alois P. Heinz_, Nov 05 2020
%t A218222 a[n_] := a[n] = If[n<2, 1, n*Sum[a[i]*a[n-i], {i, 1, n-1}]];
%t A218222 Array[a, 20] (* _Jean-François Alcover_, Dec 18 2020, after Maple *)
%o A218222 (PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x+x*deriv(A^2+x*O(x^n))); polcoeff(A, n)}
%o A218222 for(n=0, 30, print1(a(n), ", "))
%Y A218222 Cf. A000699, A112915, A185183.
%K A218222 nonn
%O A218222 1,2
%A A218222 _Paul D. Hanna_, Jan 31 2013