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 A100192 #25 Jan 16 2025 21:55:03
%S A100192 1,4,18,82,374,1704,7752,35214,159750,723880,3276908,14821668,
%T A100192 66991436,302605528,1366182276,6165204102,27811282374,125415953208,
%U A100192 565408947756,2548400193852,11483706241044,51739037228688,233070330199296,1049777052815052,4727770393417884
%N A100192 a(n) = Sum_{k=0..n} binomial(2*n,n+k)*2^k.
%C A100192 A transform of 2^n under the mapping g(x)->(1/sqrt(1-4*x))*g(x*c(x)^2), where c(x) is the g.f. of the Catalan numbers A000108. A transform of 3^n under the mapping g(x)->(1/(c(x)*sqrt(1-4*x)))*g(x*c(x)).
%C A100192 Hankel transform is A088138(n+1). - _Paul Barry_, Jan 11 2007
%H A100192 Vincenzo Librandi, <a href="/A100192/b100192.txt">Table of n, a(n) for n = 0..200</a>
%F A100192 G.f.: (sqrt(1-4*x)+1)/(sqrt(1-4*x)*(3*sqrt(1-4*x)-1)).
%F A100192 G.f.: sqrt(1-4*x)*(sqrt(1-4*x)-3*x+1)/((1-4*x)*(2-9*x)).
%F A100192 a(n) = Sum_{k=0..n} binomial(2*n, n-k)*2^k.
%F A100192 a(n) = Sum_{k=0..n} C(2*n,k)*2^(n-k). - _Paul Barry_, Jan 11 2007
%F A100192 a(n) = Sum_{k=0..n} C(n+k-1,k)*3^(n-k). - _Paul Barry_, Sep 28 2007
%F A100192 D-finite with recurrence 2*n*a(n) +(-23*n+16)*a(n-1) +3*(29*n-44)*a(n-2) +54*(-2*n+5)*a(n-3)=0. - _R. J. Mathar_, Nov 24 2012
%F A100192 a(n) ~ (9/2)^n. - _Vaclav Kotesovec_, Feb 12 2014
%F A100192 a(n) = [x^n] 1/((1 - x)^n*(1 - 3*x)). - _Ilya Gutkovskiy_, Oct 12 2017
%t A100192 CoefficientList[Series[Sqrt[1-4*x]*(Sqrt[1-4*x]-3*x+1)/((1-4*x)*(2-9*x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 12 2014 *)
%Y A100192 Cf. A032443.
%K A100192 easy,nonn
%O A100192 0,2
%A A100192 _Paul Barry_, Nov 08 2004