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 A226316 Expansion of g.f. 1/2 + 1/(1+sqrt(1-8*x+8*x^2)). 7
 1, 1, 3, 12, 56, 284, 1516, 8384, 47600, 275808, 1624352, 9694912, 58510912, 356467392, 2189331648, 13540880384, 84265071360, 527232146944, 3314742364672, 20930141861888, 132673039491072, 843959152564224, 5385800362473472, 34470606645280768, 221213787774230528, 1423139139514138624 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS From Robert A. Proctor, Jul 18 2017: (Start) a(n) is the number of words of length n on {1,2,...,r} with positive multiplicities as 1 <= r <= n avoiding the pattern 123. [This is easy to see from the next comment.] a(n) is the number of 123-avoiding ordered set partitions of {1,2,...,n}. [This is Cor. 2.3 of the Chen-Dai-Zhou reference.] (End) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..100 W. Y. C. Chen, A. Y. L. Dai and R. D. P. Zhou, Ordered Partitions Avoiding a Permutation of Length 3, arXiv preprint arXiv:1304.3187 [math.CO], 2013. Robert A. Proctor, Matthew J. Willis, Parabolic Catalan numbers count flagged Schur functions and their appearances as type A Demazure characters (key polynomials), arXiv preprint arXiv:1706.04649 [math.CO], 2017. Wikipedia, Permutation pattern FORMULA a(n) ~ sqrt((sqrt(2)-1)/Pi)*2^(n-1/2)*(2+sqrt(2))^n/n^(3/2). - Vaclav Kotesovec, Jun 29 2013 Conjecture: (n+1)*a(n) +3*(-3*n+1)*a(n-1) +4*(4*n-5)*a(n-2) +8*(-n+2)*a(n-3)=0. - R. J. Mathar, Apr 02 2015 a(n) = A000670(n) - A335515(n). - Gus Wiseman, Jun 25 2020 EXAMPLE From Gus Wiseman, Jun 25 2020: (Start) The a(0) = 1 through a(3) = 12 words that are (1,2,3)-avoiding and cover an initial interval:   ()  (1)  (1,1)  (1,1,1)            (1,2)  (1,1,2)            (2,1)  (1,2,1)                   (1,2,2)                   (1,3,2)                   (2,1,1)                   (2,1,2)                   (2,1,3)                   (2,2,1)                   (2,3,1)                   (3,1,2)                   (3,2,1) (End) MAPLE a:= proc(n) option remember; `if`(n<4, [1\$2, 3, 12][n+1],       ((9*n-3)*a(n-1) -(16*n-20)*a(n-2) +(8*n-16)*a(n-3))/(n+1))     end: seq(a(n), n=0..30);  # Alois P. Heinz, Jun 18 2013 MATHEMATICA CoefficientList[Series[1/2 + 1 / (1 + Sqrt[1 - 8 x + 8 x^2]), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 18 2013 *) allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]]; Table[Length[Select[Join@@Permutations/@allnorm[n], !MatchQ[#, {___, x_, ___, y_, ___, z_, ___}/; x

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Last modified July 8 04:16 EDT 2020. Contains 335504 sequences. (Running on oeis4.)