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%I #9 Nov 11 2016 15:40:02
%S 1,2,6,24,110,496,2156,9216,38742,160032,664532,2898848,13923468,
%T 75361600,450629592,2844358656,18224898790,116051632704,728724233988,
%U 4509502911328,27569637798116,167072272244352,1006431412676456,6037728817690112,36101656922629500
%N G.f. satisfies: A(x) = 1/A(-x*A(x)^2).
%C Compare to: W(x) = 1/W(-x*W(x)^2) when W(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
%C Compare to: B(x) = 1/B(-x*B(x)^2) when B(x) = Sum_{n>=0} (2*n)!*x^n/n!^2.
%C An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^2); for example, (*) is satisfied by G(x) = W(m*x) = LambertW(-m*x)/(-m*x) for all m, where W(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
%H Vaclav Kotesovec, <a href="/A214762/b214762.txt">Table of n, a(n) for n = 0..290</a>
%F The g.f. of this sequence is the limit of the recurrence:
%F (*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)^2))/2 starting at G_0(x) = 1+2*x.
%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 110*x^4 + 496*x^5 + 2156*x^6 +...
%e Related expansions:
%e A(x)^2 = 1 + 4*x + 16*x^2 + 72*x^3 + 352*x^4 + 1720*x^5 + 8192*x^6 +...
%e 1/A(x) = A(-x*A(x)^2) = 1 - 2*x - 2*x^2 - 8*x^3 - 34*x^4 - 112*x^5 - 324*x^6 - 896*x^7 - 1866*x^8 - 800*x^9 + 5540*x^10 +...
%o (PARI) {a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^2+x*O(x^n)))/2);polcoeff(A,n)}
%o for(n=0,31,print1(a(n),", "))
%Y Cf. A214761, A214763, A214764, A214765, A214766, A214767, A214768, A214769.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 29 2012