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A120010 G.f.: A(x) = (1-sqrt(1-4*x))/2 o x/(1-x) o (x-x^2), a composition of functions involving the Catalan function and its inverse.

%I A120010 #23 Dec 08 2021 11:26:47
%S A120010 1,1,1,2,6,18,53,158,481,1491,4688,14913,47913,155261,506881,1665643,
%T A120010 5504988,18287338,61027991,204499397,687808931,2321177071,7857504876,
%U A120010 26673769002,90783820081,309720079813,1058984020333,3628267267358
%N A120010 G.f.: A(x) = (1-sqrt(1-4*x))/2 o x/(1-x) o (x-x^2), a composition of functions involving the Catalan function and its inverse.
%C A120010 The n-th iteration of g.f. A(x) is: (1-sqrt(1-4*x))/2 o x/(1-n*x) o (x-x^2) = (1 - sqrt(1 - 4*(x-x^2)/(1-n*x+n*x^2) ))/2. See A120009 for the transpose of the composition of the same functions.
%C A120010 Row sums of A155839. [_Paul Barry_, Jan 28 2009]
%H A120010 Vincenzo Librandi, <a href="/A120010/b120010.txt">Table of n, a(n) for n = 1..200</a>
%H A120010 S. B. Ekhad and M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt"> Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017)
%F A120010 G.f.: A(x) = (1 - sqrt(1 - 4*(x-x^2)/(1-x+x^2) ))/2.
%F A120010 a(n)=sum{k=0..n, sum{j=0..n, (-1)^(n-j)*C(j+1,n-j)*C(j,k)*if(k<=j, A000108(j-k),0)}}. [offset 0]. [_Paul Barry_, Jan 28 2009]
%F A120010 Conjecture: n*a(n) +2*(4-3*n)*a(n-1) +(11*n-26)*a(n-2) +10*(3-n)*a(n-3) +5*(n-4)*a(n-4)= 0. - _R. J. Mathar_, Nov 14 2011
%F A120010 a(n) ~ sqrt(5*sqrt(5)-5) * (5+sqrt(5))^n / (sqrt(Pi) * n^(3/2) * 2^(n+7/2)). - _Vaclav Kotesovec_, Feb 13 2014
%F A120010 Equivalently, a(n) ~ 5^((n+1)/2) * phi^(n - 1/2) / (8 * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 08 2021
%e A120010 G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 18*x^6 + 53*x^7 + 158*x^8 +...
%t A120010 Rest[CoefficientList[Series[(1-Sqrt[1-4*(x-x^2)/(1-x+x^2)])/2, {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%o A120010 (PARI) {a(n)=polcoeff((1 - sqrt(1 - 4*(x-x^2)/(1-x+x^2+x*O(x^n)) ))/2,n)}
%o A120010 for(n=1,35,print1(a(n),", "))
%Y A120010 Cf. A120009 (composition transpose), A000108 (Catalan).
%K A120010 nonn
%O A120010 1,4
%A A120010 _Paul D. Hanna_, Jun 03 2006