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 A269353 #16 Nov 12 2023 13:00:36
%S A269353 1,2,10,80,910,14180,294820,7898900,262166950,10401023300,
%T A269353 480352145980,25403319156500,1524514142067820,103259722304891000,
%U A269353 7855753426977689320,667222680003329539400,62774912079259300426030,6482982674137778455277720,728040412975153836997647580
%N A269353 Expansion of 1/(1-2x/(1-3x/(1-5x/(1-7x/(1-11x/...))))) where 2,3,5,7,11,... are the prime numbers.
%H A269353 Vaclav Kotesovec, <a href="/A269353/b269353.txt">Table of n, a(n) for n = 0..300</a>
%F A269353 G.f.: 1/(1-2x/(1-3x/(1-5x/(1-7x/(1-11x/(1-13x/(...))))))), by definition.
%p A269353 b:= proc(x, y) option remember; `if`(x=0 and y=0, 1,
%p A269353 `if`(x>0, b(x-1, y)*ithprime(y-x+1), 0)+`if`(y>x, b(x, y-1), 0))
%p A269353 end:
%p A269353 a:= n-> b(n$2):
%p A269353 seq(a(n), n=0..20); # _Alois P. Heinz_, Nov 12 2023
%t A269353 CoefficientList[Series[1/(1+ContinuedFractionK[-Prime[k]*x,1,{k,1,50}]),{x,0,50}],x]
%Y A269353 Cf. A000040, A285407.
%K A269353 nonn
%O A269353 0,2
%A A269353 _Benedict W. J. Irwin_, Feb 25 2016
%E A269353 Offset corrected by _Vaclav Kotesovec_, Aug 26 2017