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A014318 Convolution of Catalan numbers and powers of 2. 10
1, 3, 8, 21, 56, 154, 440, 1309, 4048, 12958, 42712, 144210, 496432, 1735764, 6145968, 21986781, 79331232, 288307254, 1054253208, 3875769606, 14315659632, 53097586284, 197677736208, 738415086066 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Equals triangle A106270 * A000079, the powers of 2. - Gary W. Adamson, Apr 02 2009

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

a(n) = Sum(2^(n-j) * binomial(2j,j)/(j+1), j=0..n). - Emeric Deutsch, Oct 16 2008

G.f.: (1-sqrt(1-4*z))/(2*z*(1-2*z)). - Emeric Deutsch, Oct 16 2008

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

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

CROSSREFS

Cf. A000108, A097332, A106270.

Sequence in context: A094374 A008909 A006835 * A158495 A273720 A018037

Adjacent sequences:  A014315 A014316 A014317 * A014319 A014320 A014321

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified March 7 01:06 EST 2021. Contains 341853 sequences. (Running on oeis4.)