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 A126221 #8 Jul 22 2022 13:14:47 %S A126221 1,2,5,13,35,98,286,869,2739,8910,29754,101498,352222,1239332,4410204, %T A126221 15840813,57344451,208976022,765945954,2821516398,10439890026, %U A126221 38781926652,144580149924,540737349858,2028319233390,7628680720908 %N A126221 a(n)=c(n)+c(n-1)+2*c(n-2)+4*c(n-3)+8*c(n-4)+...+2^(n-2)*c(1)+2^(n-1)*c(0), where c(k) are the Catalan numbers (A000108). %C A126221 Row sums of A125177. %C A126221 Equals the eigensequence of a triangle with A000108 as the left border and the rest 1's. - _Gary W. Adamson_, Jul 24 2010 %F A126221 G.f.: (1-x)*(1-sqrt(1-4*x)) / (2*x*(1-2*x)). %F A126221 D-finite with recurrence (n+1)*a(n) +(-7*n+1)*a(n-1) +2*(7*n-8)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - _R. J. Mathar_, Jul 22 2022 %e A126221 a(4)=35 because c(4)+c(3)+2*c(2)+4*c(1)+8*c(0) = 14+5+2*2+4*1+8*1 = 35. %p A126221 c:=n->binomial(2*n,n)/(n+1): a:=n->c(n)+sum(2^(n-j-1)*c(j),j=0..n-1): seq(a(n),n=0..30); %Y A126221 Cf. A125177. %K A126221 nonn %O A126221 0,2 %A A126221 _Emeric Deutsch_, Dec 28 2006