A014318 Convolution of Catalan numbers and powers of 2.

1, 3, 8, 21, 56, 154, 440, 1309, 4048, 12958, 42712, 144210, 496432, 1735764, 6145968, 21986781, 79331232, 288307254, 1054253208, 3875769606, 14315659632, 53097586284, 197677736208, 738415086066

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)

a(n) = Sum_{j=0..n} abs(A106270(n, j)) * A000079(j). - Gary W. Adamson, Apr 02 2009

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

Cf. A000079, A000108, A087960, A097332, A106270.

Sequence in context: A094374 A008909 A006835 * A158495 A349186 A273720

Adjacent sequences: A014315 A014316 A014317 * A014319 A014320 A014321

Keyword

nonn

Author

N. J. A. Sloane

Status

approved