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

Author

N. J. A. Sloane

Keywords

• nonn

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.

Crossrefs

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

Programs

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

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 *)

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

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