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%I #6 Sep 10 2017 14:34:51
%S 2,5,12,36,81,174,413,889,1870,4031,8490,17580,36647,75801,154676,
%T 316873,646614,1309277,2653548,5358828,10786403,21697201,43539382,
%U 87208388,174392929,348359875,694913277,1384281163,2755398784,5476741024,10878139055,21590446589
%N Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with three.
%C An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.
%H Alois P. Heinz, <a href="/A292169/b292169.txt">Table of n, a(n) for n = 3..3424</a>
%e a(3) = 2: 312, 321.
%e a(4) = 5: 3124, 3142, 3214, 3241, 3421
%e a(5) = 12: 31245, 31425, 31452, 32145, 32415, 32451, 34215, 34251, 34521, 35214, 35241, 35421.
%p b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p add(b(u-j, o+j-1, j), j=1..min(t, u))+
%p add(b(u+j-1, o-j, j), j=1..min(t, o)))
%p end:
%p a:= n-> b(0, n, 3)-b(0, n, 2):
%p seq(a(n), n=3..50);
%Y Column k=3 of A291684.
%K nonn
%O 3,1
%A _Alois P. Heinz_, Sep 10 2017
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