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G.f. satisfies: A(x/A(x)) = 1 + sqrt(x - x/A(x)).
1

%I #9 Apr 26 2018 21:24:05

%S 1,1,1,3,15,102,861,8593,98453,1269924,18187062,286183564,4907331899,

%T 91082993194,1819518069135,38929958186607,888318740697313,

%U 21536467340324252,552893064959418966,14985039828839650746

%N G.f. satisfies: A(x/A(x)) = 1 + sqrt(x - x/A(x)).

%F Let G(x) be the g.f. of A185754, then g.f. A(x) satisfies:

%F (1) x + (A(x) - 1)^2 = G(x),

%F (2) x * A( G(x) ) = G(x),

%F (3) G( x/A(x) ) = x.

%e G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 15*x^4 + 102*x^5 + 861*x^6 +...

%e RELATED SERIES.

%e A(x/A(x)) = 1 + x + x^3 + 5*x^4 + 37*x^5 + 329*x^6 + 3415*x^7 + 40328*x^8 + 532749*x^9 + 7777531*x^10 + 124315519*x^11 + ...

%e x - x/A(x) = x^2 + 2*x^4 + 10*x^5 + 75*x^6 + 668*x^7 + 6929*x^8 + 81684*x^9 + 1076987*x^10 + 15694214*x^11 + 250460767*x^12 + ...

%e sqrt(x - x/A(x)) = x + x^3 + 5*x^4 + 37*x^5 + 329*x^6 + 3415*x^7 + 40328*x^8 + ...

%e The g.f. of A185754 begins:

%e G(x) = x + x^2 + 2*x^3 + 7*x^4 + 36*x^5 + 243*x^6 + 2016*x^7 +...

%e where

%e (A(x) - 1)^2 = x^2 + 2*x^3 + 7*x^4 + 36*x^5 + 243*x^6 + 2016*x^7 +...

%e A(G(x)) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 243*x^5 + 2016*x^6 +...

%o (PARI) {a(n) = local(A=x+x^2); for(i=1,n, A = 2*A - x -(x/serreverse(A + x^2*O(x^n)) - 1)^2); polcoeff(x/serreverse(A + x^2*O(x^n)), n)}

%o for(n=0,30, print1(a(n),", "))

%Y Cf. A185754.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Feb 02 2011