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T(n, k) counts Schroeder n-paths whose ascent starting at the initial vertex has length k. Triangle T(n,k), read by rows.
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%I #30 Nov 16 2023 15:55:50

%S 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,

%T 58,11,1,1806,3218,2126,978,334,82,13,1,8558,15310,10286,4942,1838,

%U 526,110,15,1,41586,74614,50746,25150,9922,3142,778,142,17,1

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

%C 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 .

%C Transpose of triangular array A033878. - _Michel Marcus_, May 02 2015

%C The triangle is the Riordan square (A321620) of A155069. - _Peter Luschny_, Feb 01 2020

%H Lili Mu and Sai-nan Zheng, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Zheng/zheng8.html">On the Total Positivity of Delannoy-Like Triangles</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.6.

%H E. Pergola and R. A. Sulanke, <a href="https://cs.uwaterloo.ca/journals/JIS/PergolaSulanke/">Schroeder Triangles, Paths and Parallelogram Polyominoes</a>, Journal of Integer Sequences, Vol. 1 (1998), #98.1.7.

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

%F T(n,0) = A155069(n). - _Philippe Deléham_, Nov 03 2009

%e Triangle begins:

%e 1;

%e 1, 1;

%e 2, 3, 1;

%e 6, 10, 5, 1;

%e 22, 38, 22, 7, 1;

%e 90, 158, 98, 38, 9, 1;

%e 394, 698, 450, 194, 58, 11, 1;

%e 1806, 3218, 2126, 978, 334, 82, 13, 1;

%e 8558, 15310, 10286, 4942, 1838, 526, 110, 15, 1;

%e 41586, 74614, 50746, 25150, 9922, 3142, 778, 142, 17, 1 ; ...

%e ...

%e The production matrix M begins:

%e 1, 1

%e 1, 2, 1

%e 1, 2, 2, 1

%e 1, 2, 2, 2, 1

%e 1, 2, 2, 2, 2, 1

%e ...

%p # The function RiordanSquare is defined in A321620.

%p RiordanSquare((3-x-sqrt(1-6*x+x^2))/2, 10); # _Peter Luschny_, Feb 01 2020

%p # Alternative:

%p A132372 := proc(dim) # dim is the number of rows requested.

%p local T, j, A, k, C, m; m := 1;

%p T := [seq([seq(0, j = 0..k)], k = 0..dim-1)];

%p A := [seq(ifelse(k = 0, 1 + x, 2 - irem(k, 2)), k = 0..dim-2)];

%p C := [seq(1, k = 1..dim+1)]; C[1] := 0;

%p for k from 0 to dim - 1 do

%p for j from k + 1 by -1 to 2 do

%p C[j] := C[j-1] + C[j+1] * A[j-1] od;

%p T[m] := [seq(coeff(C[2], x, j), j = 0..k)];

%p m := m + 1

%p od; ListTools:-Flatten(T) end:

%p A132372(10); # _Peter Luschny_, Nov 16 2023

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

%Y Cf. A155069, A321620.

%K easy,nonn,tabl

%O 0,4

%A _Philippe Deléham_, Nov 20 2007