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 A206308 #15 Dec 21 2022 20:20:09
%S A206308 1,19,799,57527,6327971,987163475,207304329751,56386777692271,
%T A206308 19284277970756683,8099396747717806859,4098294754345210270655,
%U A206308 2458976852607126162392999,1726201750530202565999885299,1401675821430524483591906862787
%N A206308 a(n) = ((2*n+2)*(2*n+3) - 1)*a(n-1) + 2*n*(2*n+1)*a(n-2), a(0)=1, a(1)=19.
%C A206308 The denominators of the fractions limiting to the value of A206530.
%D A206308 E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.
%H A206308 G. C. Greubel, <a href="/A206308/b206308.txt">Table of n, a(n) for n = 0..220</a>
%F A206308 a(n) = A125202(n+2)*a(n-1) + A002943(n)*a(n-2), with a(0) = 1, a(1) = 19.
%t A206308 RecurrenceTable[{a[n]==((2n+3)(2n+2)-1)a[n-1]+2n(2n+1)a[n-2], a[0]==1, a[1]==19}, a, {n,15}]
%o A206308 (Magma) [n le 2 select 19^(n-1) else (4*n^2+2*n-1)*Self(n-1) + 2*(n-1)*(2*n-1)*Self(n-2): n in [1..31]]; // _G. C. Greubel_, Dec 21 2022
%o A206308 (SageMath)
%o A206308 @CachedFunction # a = A206308
%o A206308 def a(n): return 19^n if (n<2) else (4*n^2+10*n+5)*a(n-1) + 2*n*(2*n+1)*a(n-2)
%o A206308 [a(n) for n in range(31)] # _G. C. Greubel_, Dec 21 2022
%Y A206308 Cf. A002943, A125202, A206307, A206530.
%K A206308 nonn,frac
%O A206308 0,2
%A A206308 _Seiichi Kirikami_, Feb 11 2012