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 A123899 #12 Jun 04 2019 10:37:28
%S A123899 1,2,3,12,60,360,252,2016,36288,362880,4989600,11975040,622702080,
%T A123899 8717829120,65383718400,5230697472000,2736057139200,49249028505600,
%U A123899 30411275102208,608225502044160,25545471085854720000
%N A123899 a(n) = (n+1)!/(d(n)*d(n+1)) where d(n) = cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms.
%H A123899 J. Sondow, <a href="https://www.jstor.org/stable/27642006">A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly 113 (2006) 637-641.
%H A123899 J. Sondow, <a href="https://arxiv.org/abs/0704.1282">A geometric proof that e is irrational and a new measure of its irrationality</a>, arXiv:0704.1282 [math.HO], 2007-2010.
%H A123899 J. Sondow and K. Schalm, <a href="http://arxiv.org/abs/0709.0671">Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II</a>, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
%F A123899 a(n) = (n+1)!/(A093101(n)*A093101(n+1)) where A093101(n)=gcd(n!,1+n+n(n-1)+...+n!).
%e A123899 a(2) = 3 because (2+1)!/(d(2)*d(3)) = 3!/(gcd(2,5)*gcd(6,16)) = 6/2 = 3.
%t A123899 (A[n_] := If[n==0,1,n*A[n-1]+1]; d[n_] := GCD[A[n],n! ]; Table[(n+1)!/(d[n]*d[n+1]), {n,0,22}])
%Y A123899 Cf. A000522, A061354, A093101, A123900, A123901.
%K A123899 easy,nonn
%O A123899 0,2
%A A123899 _Jonathan Sondow_, Oct 18 2006