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Number triangle T(n,k) = (k-(k-1)*0^(n-k))*[k<=n].
5

%I #28 Dec 17 2023 10:31:14

%S 1,0,1,0,1,1,0,1,2,1,0,1,2,3,1,0,1,2,3,4,1,0,1,2,3,4,5,1,0,1,2,3,4,5,

%T 6,1,0,1,2,3,4,5,6,7,1,0,1,2,3,4,5,6,7,8,1,0,1,2,3,4,5,6,7,8,9,1

%N Number triangle T(n,k) = (k-(k-1)*0^(n-k))*[k<=n].

%C Row sums are n*(n-1)/2+1 (essentially A000124). Diagonal sums are A114220. First difference triangle of A077028, when this is viewed as a number triangle.

%C From _R. J. Mathar_, Mar 22 2013: (Start)

%C The matrix inverse is

%C 1;

%C 0, 1;

%C 0, -1, 1;

%C 0, 1, -2, 1;

%C 0, -2, 4, -3, 1;

%C 0, 6, -12, 9, -4, 1;

%C 0, -24, 48, -36, 16, -5, 1;

%C 0, 120, -240, 180, -80, 25, -6, 1;

%C 0, -720, 1440, -1080, 480, -150, 36, -7, 1;

%C ... apparently related to A208058. (End)

%C Number of permutations of length n avoiding simultaneously the patterns 132 and 321 with k left-to-right maxima (resp., right-to-left minima). A left-to-right maximum (resp., right-to-left minimum) in a permutation p(1)p(2)...p(n) is a position i such that p(j) < p(i) for all j < i (resp., p(j) > p(i) for all j > i). - _Sergey Kitaev_, Nov 18 2023

%H Tian Han and Sergey Kitaev, <a href="https://arxiv.org/abs/2311.02974">Joint distributions of statistics over permutations avoiding two patterns of length 3</a>, arXiv:2311.02974 [math.CO], 2023.

%F G.f.: (1-x-u*x + 2u*x^2)/((1-x)(1-u*x)^2), where x records length and u records left-to-right maxima (or right-to-left minima). - _Sergey Kitaev_, Nov 18 2023

%e Triangle begins

%e 1;

%e 0, 1;

%e 0, 1, 1;

%e 0, 1, 2, 1;

%e 0, 1, 2, 3, 1;

%e 0, 1, 2, 3, 4, 1;

%e 0, 1, 2, 3, 4, 5, 1;

%e 0, 1, 2, 3, 4, 5, 6, 1;

%e 0, 1, 2, 3, 4, 5, 6, 7, 1;

%e ...

%p A114219 := proc(n,k)

%p if k < 0 or k > n then

%p 0;

%p elif n = k then

%p 1;

%p else

%p k ;

%p end if;

%p end proc: # _R. J. Mathar_, Mar 22 2013

%Y Cf. A000124, A114220, A077028.

%K easy,nonn,tabl

%O 0,9

%A _Paul Barry_, Nov 18 2005