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 A220097 #73 Feb 21 2020 07:09:32
%S A220097 1,1,6,43,352,3114,29004,280221,2782476,28221784,291138856,3045298326,
%T A220097 32222872906,344293297768,3709496350512,40256666304723,
%U A220097 439645950112788,4828214610825948,53286643424088024,590705976259292856,6574347641664629388,73433973722458186608
%N A220097 Number of words on {1,1,2,2,3,3,...,n,n} avoiding the pattern 123.
%C A220097 a(n) is the number of 123-avoiding ordered set partitions of {1,...,2n} where all blocks are of size 2.
%H A220097 Alois P. Heinz, <a href="/A220097/b220097.txt">Table of n, a(n) for n = 0..931</a> (terms n=1..25 from Lara Pudwell)
%H A220097 Ferenc Balogh, <a href="https://arxiv.org/abs/1505.01389">A generalization of Gessel's generating function to enumerate words with double or triple occurrences in each letter and without increasing subsequences of a given length</a>, preprint arXiv:1505.01389 [math.CO], 2015.
%H A220097 W. Y. C. Chen, A. Y. L. Dai and R. D. P. Zhou, <a href="https://arxiv.org/abs/1304.3187">Ordered Partitions Avoiding a Permutation of Length 3</a>, arXiv preprint arXiv:1304.3187 [math.CO], 2013.
%H A220097 Anant Godbole, Adam Goyt, Jennifer Herdan, and Lara Pudwell, <a href="https://arxiv.org/abs/1212.2530">Pattern Avoidance in Ordered Set Partitions</a>, arXiv preprint arXiv:1212.2530 [math.CO], 2012.
%H A220097 Robert A. Proctor, Matthew J. Willis, <a href="https://arxiv.org/abs/1706.03094">Convexity of tableau sets for type A Demazure characters (key polynomials), parabolic Catalan numbers</a>, arXiv preprint arXiv:1706.03094 [math.CO], 2017.
%H A220097 Robert A. Proctor, Matthew J. Willis, <a href="https://arxiv.org/abs/1706.04649">Parabolic Catalan numbers count flagged Schur functions and their appearances as type A Demazure characters (key polynomials)</a>, arXiv:1706.04649 [math.CO], 2017.
%H A220097 Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/papers/pudwell_schemes.pdf">Enumeration schemes for words avoiding permutations</a>, in Permutation Patterns (2010), S. Linton, N. Ruskuc, and V. Vatter, Eds., vol. 376 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 193-211. Cambridge: Cambridge University Press.
%H A220097 Nathaniel Shar, <a href="https://pdfs.semanticscholar.org/98e3/71b675789ed6ec4f9c9cd82e2dee9ca79399.pdf">Experimental methods in permutation patterns and bijective proof</a>, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
%H A220097 N. Shar, D. Zeilberger, <a href="https://arxiv.org/abs/1411.5052">The (Ordinary) Generating Functions Enumerating 123-Avoiding Words with r Occurrences of Each of 1, 2,..., n are Always Algebraic</a>, arXiv preprint arXiv:1411.5052 [math.CO], 2014.
%F A220097 a(n) ~ 12^n/(sqrt(Pi)*(7*n/3)^(3/2)). - _Vaclav Kotesovec_, May 22 2013
%F A220097 G.f. = sqrt( 2/(1+2*x+sqrt(1-12*x))) [Chen et al.] - _N. J. A. Sloane_, Jun 09 2013
%F A220097 Conjecture: a(n) = (2/Pi)*Integral_{t=0..1} sqrt((1 - t)/t)*(16*t^2 - 4*t)^n = Catalan(2*n)*2F1(-1-2*n,-n;1/2-2*n;1/4). - _Benedict W. J. Irwin_, Oct 05 2016
%F A220097 a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(n,k)*Catalan(n+k). - _Peter Luschny_, Aug 15 2017
%F A220097 D-finite with recurrence: 4*n*(2*n+1)*a(n) +2*(-53*n^2+63*n-16)*a(n-1) +9*(13*n^2-59*n+62)*a(n-2) +18*(n-2)*(2*n-5)*a(n-3)=0. - _R. J. Mathar_, Feb 21 2020
%e A220097 For n=2, the a(2)=6 words are 1122, 1212, 1221, 2112, 2121, 2211. For n=3, 213312 would be counted because it has no increasing subsequence of length 3, but 113223 would not be counted because it does have such an increasing subsequence.
%e A220097 For n=2, the a(2)=6 ordered set partitions are 12/34, 13/24, 14/23, 34/12, 24/13, 23/14. For n=3, 46/23/15 would be counted because there is no way to choose i from the first block, j from the second block, and k from the third block such that i<j<k, but 13/25/46 would not be counted because we may select 1, 2, and 4 as a 123 pattern.
%t A220097 Rest@ CoefficientList[Series[Sqrt[2/(1 + 2 x + Sqrt[1 - 12 x])], {x, 0, 20}], x] (* _Michael De Vlieger_, Oct 05 2016 *)
%t A220097 Table[Sum[(-1)^(n+k) Binomial[n,k]CatalanNumber[n+k], {k,0,n}], {n,1,20}] (* _Peter Luschny_, Aug 15 2017 *)
%Y A220097 Cf. A266734, A266735, A226316.
%Y A220097 Column k=2 of A267479.
%Y A220097 Row sums of A288558.
%K A220097 nonn
%O A220097 0,3
%A A220097 _Lara Pudwell_, Dec 04 2012
%E A220097 a(0)=1 prepended by _Alois P. Heinz_, Nov 15 2019