This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A014318 #50 Jan 10 2023 08:00:13
%S A014318 1,3,8,21,56,154,440,1309,4048,12958,42712,144210,496432,1735764,
%T A014318 6145968,21986781,79331232,288307254,1054253208,3875769606,
%U A014318 14315659632,53097586284,197677736208,738415086066
%N A014318 Convolution of Catalan numbers and powers of 2.
%C A014318 Binomial transform of A097332: (1, 2, 3, 5, 9, 18, 39, ...). - _Gary W. Adamson_, Aug 01 2011
%C A014318 Hankel transform is A087960. - _Wathek Chammam_, Dec 02 2011
%H A014318 Fung Lam, <a href="/A014318/b014318.txt">Table of n, a(n) for n = 0..1600</a>
%H A014318 W. Chammam, F. Marcellán and R. Sfaxi, <a href="http://dx.doi.org/10.1016/j.laa.2011.10.010">Orthogonal polynomials, Catalan numbers, and a general Hankel determinant evaluation</a>, Linear Algebra and its Applications, Volume 436, Issue 7, 1 April 2012, Pages 2105-2116.
%F A014318 From _Emeric Deutsch_, Oct 16 2008: (Start)
%F A014318 G.f.: (1-sqrt(1-4*z))/(2*z*(1-2*z)).
%F A014318 a(n) = Sum_{j=0..n} (2^(n-j) * binomial(2*j,j)/(j+1)). (End)
%F A014318 a(n) = Sum_{j=0..n} abs(A106270(n, j)) * A000079(j). - _Gary W. Adamson_, Apr 02 2009
%F A014318 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
%F A014318 Asymptotics: a(n) ~ 2^(2n+1)/n^(3/2)/sqrt(Pi). - _Fung Lam_, Mar 21 2014
%F A014318 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x) * A(x)^2. - _Ilya Gutkovskiy_, Nov 21 2021
%p A014318 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
%t A014318 a[n_]:= a[n]= Sum[2^(n-j)*CatalanNumber[j], {j,0,n}];
%t A014318 Table[a[n], {n,0,40}] (* _G. C. Greubel_, Jan 09 2023 *)
%o A014318 (Magma)
%o A014318 A014318:= func< n | (&+[2^(n-j)*Catalan(j): j in [0..n]]) >;
%o A014318 [A014318(n): n in [0..40]]; // _G. C. Greubel_, Jan 09 2023
%o A014318 (SageMath)
%o A014318 def A014318(n): return sum(2^(n-j)*catalan_number(j) for j in range(n+1))
%o A014318 [A014318(n) for n in range(41)] # _G. C. Greubel_, Jan 09 2023
%Y A014318 Cf. A000079, A000108, A087960, A097332, A106270.
%K A014318 nonn
%O A014318 0,2
%A A014318 _N. J. A. Sloane_