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

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, 16, 1, 1, 247, 825, 955, 515, 141, 19, 1, 1, 502, 2116, 3178, 2310, 906, 196, 22, 1, 1, 1013, 5200, 9740, 9078, 4746, 1456, 260, 25, 1

Offset: 0

Author

Peter Luschny, Feb 03 2020

Keywords

• nonn
• tabl

Comments

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.

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

Examples

			Triangle starts:
[0] [1]
[1] [1,    1]
[2] [1,    4,    1]
[3] [1,   11,    7,    1]
[4] [1,   26,   30,   10,    1]
[5] [1,   57,  102,   58,   13,    1]
[6] [1,  120,  303,  256,   95,   16,    1]
[7] [1,  247,  825,  955,  515,  141,   19,   1]
[8] [1,  502, 2116, 3178, 2310,  906,  196,  22,  1]
[9] [1, 1013, 5200, 9740, 9078, 4746, 1456, 260, 25, 1]
...
Seen as a square array (the triangle is formed by descending antidiagonals):
1,  1,   1,    1,    1,     1,      1,      1,       1, ... [A000012]
1,  4,  11,   26,   57,   120,    247,    502,    1013, ... [A000295]
1,  7,  30,  102,  303,   825,   2116,   5200,   12381, ... [A045889]
1, 10,  58,  256,  955,  3178,   9740,  28064,   77093, ... [A055583]
1, 13,  95,  515, 2310,  9078,  32354, 106970,  333295, ...
1, 16, 141,  906, 4746, 21504,  87374, 326084, 1136799, ...
1, 19, 196, 1456, 8722, 44758, 204204, 849180, 3275931, ...
		

Links

• Donghyun Kim and Lauren Williams, Schubert polynomials and the inhomogeneous TASEP on a ring, arXiv:2102.00560 [math.CO], 2021.

Crossrefs

Row sums A006012, alternating row sums A118434 with different signs, central column A091527.

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.

Cf. A172094 (inverse up to sign).

Programs

Maple

gf := k -> 1/(((1-2*x)^k)*(1-x)^(k+1)): ser := k -> series(gf(k), x, 32):
# Prints the triangle:
seq(lprint(seq(coeff(ser(k), x, n-k), k=0..n)), n=0..6);
# Prints the square array:
seq(lprint(seq(coeff(ser(k), x, n), n=0..8)), k=0..6);

Mathematica

(* The function RiordanSquare is defined in A321620; returns the triangle as a lower triangular matrix. *)
M := RiordanSquare[(1 + x - Sqrt[1 - 6 x + x^2])/(4 x), 9];
Abs[#] & /@ Inverse[PadRight[M]]