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 A120786 #14 Jan 25 2025 09:13:47
%S A120786 1,21,211,1689,84457,1689161,16891643,1351331869,2702663881,
%T A120786 270266390531,2702663909509,108106556409753,1081065564149533,
%U A120786 4324262256635277,43242622566419631,6918819610629079929
%N A120786 Numerators of partial sums of Catalan numbers scaled by powers of 1/20.
%C A120786 Denominators are given under A120787.
%C A120786 From the expansion of 2*sqrt(5)/5 = sqrt(1-1/5) = 1-(1/10)*Sum_{k>=0} C(k)/20^k one has r := lim_{n->oo} r(n) = 2*(5 - 2*sqrt(5)) = 2*(7 - 4*phi) = 1.055728090..., where phi := (1+sqrt(5))/2 (golden section) and the partial sums r(n) are defined below.
%C A120786 This is the second member (p=1) in the second p-family of partial sums of the normalized scaled Catalan series CsnII(p) := Sum_{k>=0} C(k)/((5^k)*F(2*p+1)^(2*k)) with limit F(2*p+1)*(L(2*p+2) - L(2*p+1)*phi), with C(n) = A000108(n) (Catalan), F(n) = A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi := (1+sqrt(5))/2 (golden section).
%C A120786 The partial sums of the above mentioned second p-family are rII(p;n) := Sum_{k=0..n} C(k)/((5^k)*F(2*p+1)^(2*k)), n>=0, for p=0,1,...
%C A120786 For more details about this p-family and the other three ones see the W. Lang link under A120996.
%H A120786 Wolfdieter Lang, <a href="/A120786/a120786.txt">Rationals r(n) and limit.</a>
%F A120786 a(n) = numerator(r(n)), with the rationals r(n) := Sum_{k=0..n} C(k)/20^k with C(k) := A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.
%e A120786 Rationals r(n): [1, 21/20, 211/200, 1689/1600, 84457/80000, 1689161/1600000, 16891643/16000000, 1351331869/1280000000,...].
%K A120786 nonn,easy,frac
%O A120786 0,2
%A A120786 _Wolfdieter Lang_, Jul 20 2006