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%I #34 Oct 14 2019 11:30:32
%S 1,1,2,9,50,311,2072,14460,104346,772255,5829538,44710705,347424376,
%T 2729299748,21640457360,172957598120,1391926695402,11270059892943,
%U 91740990170150,750364940281275,6163650579487170,50824871829196575
%N G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^2 - 1)^n.
%C Compare to a g.f. of the Catalan numbers: C(x) = 1 + x*Sum_{n>=0} (C(x) - 1)^n.
%H G. C. Greubel, <a href="/A192945/b192945.txt">Table of n, a(n) for n = 0..1000</a>
%H Elżbieta Liszewska, Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%H Thomas M. Richardson, <a href="http://arxiv.org/abs/1609.01193">The three 'R's and the Riordan dual</a>, arXiv:1609.01193 [math.CO], 2016.
%F G.f. A(x) equals the formal inverse of function (x-1)*(2-x^2).
%F G.f. satisfies: A(x) = 1 + x/(2 - A(x)^2).
%F G.f.: A(x) = 1 + Series_Reversion( 2*x - x*(1+x)^2 ).
%F a(n) = (1/n)*Sum_{k=0..n-1} binomial(n+k-1, n-1)*Sum_{i=ceiling((n-k-1)/2)..n-k-1} binomial(i, n-k-i-1)*binomial(n+k+i-1, n+k-1), n > 0, a(0)=1. - _Vladimir Kruchinin_, Oct 11 2011
%F Recurrence: 8*(n-1)*n*a(n) = 34*(n-1)*(2*n-3)*a(n-1) + 3*(3*n-7)*(3*n-5)*a(n-2). - _Vaclav Kotesovec_, Nov 20 2012
%F a(n) ~ 1/3*sqrt(7/6 - 17/(6*sqrt(7)))*((17 + 7*sqrt(7))/4)^n/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Nov 20 2012
%e G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 50*x^4 + 311*x^5 + 2072*x^6 + ...
%e where (A(x) - 1)*(2 - A(x)^2) = x
%e and A(x - 2*x^2 - x^3) = 1 + x.
%e Related expansions:
%e (A(x)^2-1) = 2*x + 5*x^2 + 22*x^3 + 122*x^4 + 758*x^5 + 5047*x^6 + ...
%e (A(x)^2-1)^2 = 4*x^2 + 20*x^3 + 113*x^4 + 708*x^5 + 4736*x^6 + ...
%e (A(x)^2-1)^3 = 8*x^3 + 60*x^4 + 414*x^5 + 2909*x^6 + 20970*x^7 + ...
%e (A(x)^2-1)^4 = 16*x^4 + 160*x^5 + 1304*x^6 + 10184*x^7 + ...
%e Also,
%e A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 122*x^4 + 758*x^5 + 5047*x^6 + ...
%e A(x)^3 = 1 + 3*x + 9*x^2 + 40*x^3 + 222*x^4 + 1380*x^5 + 9191*x^6 + ...
%e where 2 + x = 2*A(x) + A(x)^2 - A(x)^3.
%t Flatten[{1,Table[1/n*Sum[Binomial[n+k-1,n-1]*Sum[Binomial[i,n-k-i-1]*Binomial[n+k+i-1,n+k-1],{i,Floor[(n-k-1)/2], n-k-1}],{k,0,n-1}],{n,1,20}]}] (* _Vaclav Kotesovec_ after _Vladimir Kruchinin_, Nov 20 2012 *)
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*sum(m=0,n,(A^2-1+x*O(x^n))^m));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+serreverse(2*x-x*(1+x)^2+x^2*O(x^n)));polcoeff(A,n)}
%o (Maxima) a(n):=if n=0 then 1 else 1/n*sum(binomial(n+k-1,n-1) *sum(binomial(i,n-k-i-1)*binomial(n+k+i-1,n+k-1),i,ceiling((n-k-1)/2), n-k-1),k,0,n-1); /* _Vladimir Kruchinin_, Oct 11 2011 */
%Y Cf. A192946, A192947, A192948.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 13 2011
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