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A098157 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n+1,2k).

%I A098157 #28 Aug 29 2024 23:33:51
%S A098157 1,1,1,0,3,1,0,1,6,1,0,0,5,10,1,0,0,1,15,15,1,0,0,0,7,35,21,1,0,0,0,1,
%T A098157 28,70,28,1,0,0,0,0,9,84,126,36,1,0,0,0,0,1,45,210,210,45,1,0,0,0,0,0,
%U A098157 11,165,462,330,55,1,0,0,0,0,0,1,66,495,924,495,66,1,0,0,0,0,0,0,13,286,1287,1716,715,78,1
%N A098157 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n+1,2k).
%C A098157 Row sums are A000079. Diagonal sums are A062200. Inverse is A065547, less the first column.
%C A098157 Number of permutations of length n avoiding simultaneously the patterns 123 and 132 with k descents. A descent in a permutation a(1)a(2)...a(n) is position i such that a(i)>a(i+1). - _Tian Han_, Nov 16 2023
%H A098157 T. Han and S. 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 A098157 T(n, k) = binomial(n+1, 2(n-k)) with 0 <= k <= n.
%F A098157 G.f.: (1 + x - q*x)/(1 - 2*q*x - q*x^2 + q^2*x^2). - _Tian Han_, Nov 16 2023
%e A098157 Rows begin:
%e A098157  {1},
%e A098157  {1,1},
%e A098157  {0,3,1},
%e A098157  {0,1,6,1},
%e A098157  {0,0,5,10,1},
%e A098157  {0,0,1,15,15,1},
%e A098157   ...
%t A098157 Table[Binomial[n+1, 2(n-k)],{n,0,11},{k,0,n}]//Flatten (* _Stefano Spezia_, Nov 16 2023 *)
%Y A098157 Cf. A000079, A062200, A065547.
%K A098157 easy,nonn,tabl
%O A098157 0,5
%A A098157 _Paul Barry_, Aug 29 2004