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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.
5
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
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).
Sequence in context: A110189 A187914 A321625 * A103136 A155856 A086960
KEYWORD
easy,nonn,tabl
AUTHOR
Philippe Deléham, Nov 20 2007
STATUS
approved