A370539 - OEIS

%I #14 Mar 14 2024 15:21:22

%S 1,2,8,30,118,462,1824,7208,28558,113274,449848,1787968,7111716,

%T 28303548,112700032,448939744,1788990454,7131191202,28433681832,

%U 113398298336,452345641820,1804739556100,7201621713568,28741559322464,114722405784428,457971605148996,1828422022584176

%N 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 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

%F (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 (2) A(x) = (1 + sqrt(1 - 4*x)) * (2-3*x + x*sqrt(1 - 4*x^2)) / (4*(1-4*x)).

%F a(n) ~ (10 + sqrt(3)) * 2^(2*n - 5). - _Vaclav Kotesovec_, Mar 14 2024

%e 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 RELATED SERIES.

%e The Catalan function C(x) = (1 - sqrt(1-4*x))/(2*x) begins

%e 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 (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 polcoeff(A,n);}

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

%o (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 polcoeff(A,n);}

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

%Y Cf. A000108, A370540.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 12 2024