login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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