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%I #16 Jul 19 2021 05:13:44
%S 1,1,3,10,36,139,560,2328,9914,43027,189619,846267,3817105,17373048,
%T 79687447,367991891,1709477714,7983062151,37454903501,176470241003,
%U 834601583199,3960757007408,18855383609076,90019104197240
%N G.f. satisfies: A(x) = Catalan(x + x^2 + x^3*A(x)) where Catalan(x) = (1-sqrt(1-4*x))/(2x) is the g.f. of A000108.
%H Vaclav Kotesovec, <a href="/A162162/b162162.txt">Table of n, a(n) for n = 0..500</a>
%F O.g.f.: A(x) = 1 + (x+x^2)*A(x)^2 + x^3*A(x)^3 [From _Simon Plouffe_].
%F a(n) = Sum_{k=0..n} Sum_{j=0..k} C(2*n-2*k+j+1,n-k)/(2*n-2*k+j+m) * C(n-k,k-j)*C(k-j,j).
%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
%F a(n,m) = Sum_{k=0..n} Sum_{j=0..k} C(2*n-2*k+j+m,n-k)*m/(2*n-2*k+j+m) * C(n-k,k-j)*C(k-j,j).
%F a(n) ~ s * sqrt((1 - r*s)/(Pi*(1 - r*s*(1 + r)))) / (2*n^(3/2)*r^n), where r = (1 + 5*sqrt(5))/62 = 0.1964570949596604593878365861880061480194047064202... and s = 4*sqrt(5)-7 = 1.944271909999158785636694674925104941762473438446... are real roots of the system of equations 1 + r*(1 + r)*s^2 + r^3*s^3 = s, r*s*(2 + 2*r + 3*r^2*s) = 1. - _Vaclav Kotesovec_, Nov 22 2017, updated Jul 19 2021
%e G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 36*x^4 + 139*x^5 + 560*x^6 +...
%e A(x) = Catalan(x + x^2 + x^3*A(x)) where:
%e Catalan(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
%t m = 24; Catala[x_] = (1 - Sqrt[1 - 4 x])/(2 x); A[_] = 0;
%t Do[A[x_] = Catala[x + x^2 + x^3 A[x]] + O[x]^m, {m}];
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Oct 22 2019 *)
%o (PARI) {a(n,m=1)=sum(k=0,n,sum(j=0,k,binomial(2*n-2*k+j+m,n-k)*m/(2*n-2*k+j+m)*binomial(n-k,k-j)*binomial(k-j,j)))}
%o (PARI) {a(n, m=1)=local(A=1+x+x*O(x^n));for(i=1,n,A=2/(1+sqrt(1-4*(x+x^2 +x^3*A))));polcoeff(A^m, n)}
%o (PARI) {a(n,m=1)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+(x+x^2)*A^2+x^3*A^3);polcoeff(A^m,n)}
%Y Cf. A000108.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 26 2009
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