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).

1, 2, 8, 30, 118, 462, 1824, 7208, 28558, 113274, 449848, 1787968, 7111716, 28303548, 112700032, 448939744, 1788990454, 7131191202, 28433681832, 113398298336, 452345641820, 1804739556100, 7201621713568, 28741559322464, 114722405784428, 457971605148996, 1828422022584176

Offset: 0

Author

Paul D. Hanna, Mar 12 2024

Keywords

• nonn

Examples

			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 + ...
RELATED SERIES.
The Catalan function C(x) = (1 - sqrt(1-4*x))/(2*x) begins
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 + ...
		

Crossrefs

Cf. A000108, A370540.

Programs

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) );
polcoeff(A,n);}
for(n=0,30, print1(a(n),", "))

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)) );
polcoeff(A,n);}
for(n=0,30, print1(a(n),", "))

Formula

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

(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).

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

a(n) ~ (10 + sqrt(3)) * 2^(2*n - 5). - Vaclav Kotesovec, Mar 14 2024