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%I #12 Feb 23 2016 11:47:10
%S 0,0,1,43,1879,102011,7235651,674641325,81537026047,12498099730471,
%T 2375632826877259,548818073236649129,151476182218777630655,
%U 49229890784448694885163,18608906461974462064310179,8094874797394331233877338741,4015057931973886657462193434111
%N Number of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length < n.
%C Or number of words on {1,1,2,2,...,n,n} avoiding the pattern 12...n.
%H Alois P. Heinz, <a href="/A267532/b267532.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = (2*n)! * ( 1/(2^n) - Sum_{k=0..n} (-1)^k * C(n,k) / (n+k)! ).
%F a(n) = A000680(n) - A006902(n).
%F a(n) = A267479(n,n-1) for n>0.
%F a(n) = Sum_{k=0..n-1} A267480(n,k).
%e a(2) = 1: 2211.
%e a(3) = 43: 113322, 131322, 133122, 133212, 133221, 211332, 213132, 213312, 213321, 221133, 221313, 221331, 223113, 223131, 223311, 231132, 231312, 231321, 232113, 232131, 232311, 233112, 233121, 233211, 311322, 313122, 313212, 313221, 321132, 321312, 321321, 322113, 322131, 322311, 323112, 323121, 323211, 331122, 331212, 331221, 332112, 332121, 332211.
%p b:= proc(n) option remember; `if`(n<3, [1$2, 5][n+1], (
%p (n^3+n^2-7*n+4)*b(n-1)-2*(2*n-3)*(n-1)^3*b(n-2))/(n-2))
%p end:
%p a:= n-> (2*n)!/(2^n)-b(n):
%p seq(a(n), n=0..20);
%Y Cf. A000079, A000142, A000680, A006902, A010050, A267479, A267480, A269042.
%Y Column k=2 of A269129.
%K nonn
%O 0,4
%A _Alois P. Heinz_, Jan 16 2016
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