A132372 T(n, k) counts Schroeder n-paths whose ascent starting at the initial vertex has length k. Triangle T(n,k), read by rows.

1, 1, 1, 2, 3, 1, 6, 10, 5, 1, 22, 38, 22, 7, 1, 90, 158, 98, 38, 9, 1, 394, 698, 450, 194, 58, 11, 1, 1806, 3218, 2126, 978, 334, 82, 13, 1, 8558, 15310, 10286, 4942, 1838, 526, 110, 15, 1, 41586, 74614, 50746, 25150, 9922, 3142, 778, 142, 17, 1

Offset: 0

Author

Philippe Deléham, Nov 20 2007

Keywords

• easy
• nonn
• tabl

Comments

Triangle T(n,k), 0<=k<=n, read by rows given by [1,1,2,1,2,1,2,1,2,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938 .

Transpose of triangular array A033878. - Michel Marcus, May 02 2015

The triangle is the Riordan square (A321620) of A155069. - Peter Luschny, Feb 01 2020

Examples

			Triangle begins:
      1;
      1,     1;
      2,     3,     1;
      6,    10,     5,     1;
     22,    38,    22,     7,    1;
     90,   158,    98,    38,    9,    1;
    394,   698,   450,   194,   58,   11,   1;
   1806,  3218,  2126,   978,  334,   82,  13,   1;
   8558, 15310, 10286,  4942, 1838,  526, 110,  15,  1;
  41586, 74614, 50746, 25150, 9922, 3142, 778, 142, 17, 1 ; ...
...
The production matrix M begins:
  1, 1
  1, 2, 1
  1, 2, 2, 1
  1, 2, 2, 2, 1
  1, 2, 2, 2, 2, 1
  ...
		

Links

• Lili Mu and Sai-nan Zheng, On the Total Positivity of Delannoy-Like Triangles, Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.6.
• E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, Journal of Integer Sequences, Vol. 1 (1998), #98.1.7.

Crossrefs

Cf. A006318, A103136 (signed version), A033878 (transpose).

Cf. A155069, A321620.

Programs

Maple

# The function RiordanSquare is defined in A321620.
RiordanSquare((3-x-sqrt(1-6*x+x^2))/2, 10); # Peter Luschny, Feb 01 2020
# Alternative:
A132372 := proc(dim) # dim is the number of rows requested.
local T, j, A, k, C, m; m := 1;
T := [seq([seq(0, j = 0..k)], k = 0..dim-1)];
A := [seq(ifelse(k = 0, 1 + x, 2 - irem(k, 2)), k = 0..dim-2)];
C := [seq(1, k = 1..dim+1)]; C[1] := 0;
for k from 0 to dim - 1 do
    for j from k + 1 by -1 to 2 do
        C[j] := C[j-1] + C[j+1] * A[j-1] od;
    T[m] := [seq(coeff(C[2], x, j), j = 0..k)];
    m := m + 1
od; ListTools:-Flatten(T) end:
A132372(10);  # Peter Luschny, Nov 16 2023

Formula

Sum_{k, 0<=k<=n} T(n,k) = A006318(n) .

T(n,0) = A155069(n). - Philippe Deléham, Nov 03 2009