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 A114219 #28 Dec 17 2023 10:31:14 %S A114219 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 A114219 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 A114219 Number triangle T(n,k) = (k-(k-1)*0^(n-k))*[k<=n]. %C A114219 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 A114219 From _R. J. Mathar_, Mar 22 2013: (Start) %C A114219 The matrix inverse is %C A114219 1; %C A114219 0, 1; %C A114219 0, -1, 1; %C A114219 0, 1, -2, 1; %C A114219 0, -2, 4, -3, 1; %C A114219 0, 6, -12, 9, -4, 1; %C A114219 0, -24, 48, -36, 16, -5, 1; %C A114219 0, 120, -240, 180, -80, 25, -6, 1; %C A114219 0, -720, 1440, -1080, 480, -150, 36, -7, 1; %C A114219 ... apparently related to A208058. (End) %C A114219 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 A114219 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 A114219 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 A114219 Triangle begins %e A114219 1; %e A114219 0, 1; %e A114219 0, 1, 1; %e A114219 0, 1, 2, 1; %e A114219 0, 1, 2, 3, 1; %e A114219 0, 1, 2, 3, 4, 1; %e A114219 0, 1, 2, 3, 4, 5, 1; %e A114219 0, 1, 2, 3, 4, 5, 6, 1; %e A114219 0, 1, 2, 3, 4, 5, 6, 7, 1; %e A114219 ... %p A114219 A114219 := proc(n,k) %p A114219 if k < 0 or k > n then %p A114219 0; %p A114219 elif n = k then %p A114219 1; %p A114219 else %p A114219 k ; %p A114219 end if; %p A114219 end proc: # _R. J. Mathar_, Mar 22 2013 %Y A114219 Cf. A000124, A114220, A077028. %K A114219 easy,nonn,tabl %O A114219 0,9 %A A114219 _Paul Barry_, Nov 18 2005