OFFSET
0,2
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
EXAMPLE
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 + ...
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 12 2024
STATUS
approved