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A370539 Expansion of g.f. (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x), where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

%I A370539 #14 Mar 14 2024 15:21:22
%S A370539 1,2,8,30,118,462,1824,7208,28558,113274,449848,1787968,7111716,
%T A370539 28303548,112700032,448939744,1788990454,7131191202,28433681832,
%U A370539 113398298336,452345641820,1804739556100,7201621713568,28741559322464,114722405784428,457971605148996,1828422022584176
%N A370539 Expansion of g.f. (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x), where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).
%F A370539 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A370539 (1) A(x) = (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x), where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).
%F A370539 (2) A(x) = (1 + sqrt(1 - 4*x)) * (2-3*x + x*sqrt(1 - 4*x^2)) / (4*(1-4*x)).
%F A370539 a(n) ~ (10 + sqrt(3)) * 2^(2*n - 5). - _Vaclav Kotesovec_, Mar 14 2024
%e A370539 G.f.: A(x) = 1 + 2*x + 8*x^2 + 30*x^3 + 118*x^4 + 462*x^5 + 1824*x^6 + 7208*x^7 + 28558*x^8 + 113274*x^9 + 449848*x^10 + ...
%e A370539 RELATED SERIES.
%e A370539 The Catalan function C(x) = (1 - sqrt(1-4*x))/(2*x) begins
%e A370539 C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + ... + A000108(n)*x^n + ...
%o A370539 (PARI) {a(n) = my(x = 'x + O('x^(n+3)), C(x) = (1 - sqrt(1 - 4*x))/(2*x), A = (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) );
%o A370539 polcoeff(A,n);}
%o A370539 for(n=0,30, print1(a(n),", "))
%o A370539 (PARI) {a(n) = my(x = 'x + O('x^(n+3)), A = (1 + sqrt(1 - 4*x)) * sqrt( (1 - 2*x)*(1 - sqrt(1 - 4*x^2))/2 ) / (2*x*(1-4*x)) );
%o A370539 polcoeff(A,n);}
%o A370539 for(n=0,30, print1(a(n),", "))
%Y A370539 Cf. A000108, A370540.
%K A370539 nonn
%O A370539 0,2
%A A370539 _Paul D. Hanna_, Mar 12 2024