OFFSET
0,2
COMMENTS
Binomial transform of A097332: (1, 2, 3, 5, 9, 18, 39, ...). - Gary W. Adamson, Aug 01 2011
Hankel transform is A087960. - Wathek Chammam, Dec 02 2011
LINKS
Fung Lam, Table of n, a(n) for n = 0..1600
W. Chammam, F. Marcellán and R. Sfaxi, Orthogonal polynomials, Catalan numbers, and a general Hankel determinant evaluation, Linear Algebra and its Applications, Volume 436, Issue 7, 1 April 2012, Pages 2105-2116.
FORMULA
From Emeric Deutsch, Oct 16 2008: (Start)
G.f.: (1-sqrt(1-4*z))/(2*z*(1-2*z)).
a(n) = Sum_{j=0..n} (2^(n-j) * binomial(2*j,j)/(j+1)). (End)
Recurrence: (n+1)*a(n) = 32*(2*n-7)*a(n-5) + 48*(8-3*n)*a(n-4) + 8*(16*n-29)*a(n-3) + 4*(13-14*n)*a(n-2) + 12*n*a(n-1), n>=5. - Fung Lam, Mar 09 2014
Asymptotics: a(n) ~ 2^(2n+1)/n^(3/2)/sqrt(Pi). - Fung Lam, Mar 21 2014
G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x) * A(x)^2. - Ilya Gutkovskiy, Nov 21 2021
MAPLE
a:=proc(n) options operator, arrow: sum(2^(n-j)*binomial(2*j, j)/(j+1), j=0..n) end proc: seq(a(n), n=0..23); # Emeric Deutsch, Oct 16 2008
MATHEMATICA
a[n_]:= a[n]= Sum[2^(n-j)*CatalanNumber[j], {j, 0, n}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jan 09 2023 *)
PROG
(Magma)
A014318:= func< n | (&+[2^(n-j)*Catalan(j): j in [0..n]]) >;
[A014318(n): n in [0..40]]; // G. C. Greubel, Jan 09 2023
(SageMath)
def A014318(n): return sum(2^(n-j)*catalan_number(j) for j in range(n+1))
[A014318(n) for n in range(41)] # G. C. Greubel, Jan 09 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved