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A331969 T(n, k) = [x^(n-k)] 1/(((1 - 2*x)^k)*(1 - x)^(k + 1)). Triangle read by rows, for 0 <= k <= n.

%I A331969 #28 Aug 27 2021 16:29:10
%S A331969 1,1,1,1,4,1,1,11,7,1,1,26,30,10,1,1,57,102,58,13,1,1,120,303,256,95,
%T A331969 16,1,1,247,825,955,515,141,19,1,1,502,2116,3178,2310,906,196,22,1,1,
%U A331969 1013,5200,9740,9078,4746,1456,260,25,1
%N A331969 T(n, k) = [x^(n-k)] 1/(((1 - 2*x)^k)*(1 - x)^(k + 1)). Triangle read by rows, for 0 <= k <= n.
%C A331969 The triangle is the matrix inverse of the Riordan square (see A321620) generated by (1 + x - sqrt(1 - 6*x + x^2))/(4*x) (see A172094), where we take the absolute value of the terms.
%C A331969 T(n,k) is the number of evil-avoiding (2413, 3214, 4132, and 4213 avoiding) permutations of length (n+2) that start with 1 and whose inverse has k descents. - _Donghyun Kim_, Aug 16 2021
%H A331969 Donghyun Kim and Lauren Williams, <a href="https://arxiv.org/abs/2102.00560">Schubert polynomials and the inhomogeneous TASEP on a ring</a>, arXiv:2102.00560 [math.CO], 2021.
%e A331969 Triangle starts:
%e A331969 [0] [1]
%e A331969 [1] [1,    1]
%e A331969 [2] [1,    4,    1]
%e A331969 [3] [1,   11,    7,    1]
%e A331969 [4] [1,   26,   30,   10,    1]
%e A331969 [5] [1,   57,  102,   58,   13,    1]
%e A331969 [6] [1,  120,  303,  256,   95,   16,    1]
%e A331969 [7] [1,  247,  825,  955,  515,  141,   19,   1]
%e A331969 [8] [1,  502, 2116, 3178, 2310,  906,  196,  22,  1]
%e A331969 [9] [1, 1013, 5200, 9740, 9078, 4746, 1456, 260, 25, 1]
%e A331969 ...
%e A331969 Seen as a square array (the triangle is formed by descending antidiagonals):
%e A331969 1,  1,   1,    1,    1,     1,      1,      1,       1, ... [A000012]
%e A331969 1,  4,  11,   26,   57,   120,    247,    502,    1013, ... [A000295]
%e A331969 1,  7,  30,  102,  303,   825,   2116,   5200,   12381, ... [A045889]
%e A331969 1, 10,  58,  256,  955,  3178,   9740,  28064,   77093, ... [A055583]
%e A331969 1, 13,  95,  515, 2310,  9078,  32354, 106970,  333295, ...
%e A331969 1, 16, 141,  906, 4746, 21504,  87374, 326084, 1136799, ...
%e A331969 1, 19, 196, 1456, 8722, 44758, 204204, 849180, 3275931, ...
%p A331969 gf := k -> 1/(((1-2*x)^k)*(1-x)^(k+1)): ser := k -> series(gf(k), x, 32):
%p A331969 # Prints the triangle:
%p A331969 seq(lprint(seq(coeff(ser(k), x, n-k), k=0..n)), n=0..6);
%p A331969 # Prints the square array:
%p A331969 seq(lprint(seq(coeff(ser(k), x, n), n=0..8)), k=0..6);
%t A331969 (* The function RiordanSquare is defined in A321620; returns the triangle as a lower triangular matrix. *)
%t A331969 M := RiordanSquare[(1 + x - Sqrt[1 - 6 x + x^2])/(4 x), 9];
%t A331969 Abs[#] & /@ Inverse[PadRight[M]]
%Y A331969 Row sums A006012, alternating row sums A118434 with different signs, central column A091527.
%Y A331969 T(n, 1) = A000295(n+1) for n >= 1, T(n, 2) = A045889(n-2) for n >= 2, T(n, 3) = A055583(n-3) for n >= 3.
%Y A331969 Cf. A172094 (inverse up to sign).
%K A331969 nonn,tabl
%O A331969 0,5
%A A331969 _Peter Luschny_, Feb 03 2020