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

0,4

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

Links

Table of n, a(n) for n=0..54.

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.

Formula

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

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

Example

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
  ...

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

Crossrefs

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

Cf. A155069, A321620.

Sequence in context: A110189 A187914 A321625 * A103136 A155856 A086960

Adjacent sequences: A132369 A132370 A132371 * A132373 A132374 A132375

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Nov 20 2007

Status

approved