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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.

%I A132372 #30 Nov 16 2023 15:55:50
%S A132372 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 A132372 58,11,1,1806,3218,2126,978,334,82,13,1,8558,15310,10286,4942,1838,
%U A132372 526,110,15,1,41586,74614,50746,25150,9922,3142,778,142,17,1
%N 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.
%C A132372 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 A132372 Transpose of triangular array A033878. - _Michel Marcus_, May 02 2015
%C A132372 The triangle is the Riordan square (A321620) of A155069. - _Peter Luschny_, Feb 01 2020
%H A132372 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 A132372 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 A132372 Sum_{k, 0<=k<=n} T(n,k) = A006318(n) .
%F A132372 T(n,0) = A155069(n). - _Philippe Deléham_, Nov 03 2009
%e A132372 Triangle begins:
%e A132372       1;
%e A132372       1,     1;
%e A132372       2,     3,     1;
%e A132372       6,    10,     5,     1;
%e A132372      22,    38,    22,     7,    1;
%e A132372      90,   158,    98,    38,    9,    1;
%e A132372     394,   698,   450,   194,   58,   11,   1;
%e A132372    1806,  3218,  2126,   978,  334,   82,  13,   1;
%e A132372    8558, 15310, 10286,  4942, 1838,  526, 110,  15,  1;
%e A132372   41586, 74614, 50746, 25150, 9922, 3142, 778, 142, 17, 1 ; ...
%e A132372 ...
%e A132372 The production matrix M begins:
%e A132372   1, 1
%e A132372   1, 2, 1
%e A132372   1, 2, 2, 1
%e A132372   1, 2, 2, 2, 1
%e A132372   1, 2, 2, 2, 2, 1
%e A132372   ...
%p A132372 # The function RiordanSquare is defined in A321620.
%p A132372 RiordanSquare((3-x-sqrt(1-6*x+x^2))/2, 10); # _Peter Luschny_, Feb 01 2020
%p A132372 # Alternative:
%p A132372 A132372 := proc(dim) # dim is the number of rows requested.
%p A132372 local T, j, A, k, C, m; m := 1;
%p A132372 T := [seq([seq(0, j = 0..k)], k = 0..dim-1)];
%p A132372 A := [seq(ifelse(k = 0, 1 + x, 2 - irem(k, 2)), k = 0..dim-2)];
%p A132372 C := [seq(1, k = 1..dim+1)]; C[1] := 0;
%p A132372 for k from 0 to dim - 1 do
%p A132372     for j from k + 1 by -1 to 2 do
%p A132372         C[j] := C[j-1] + C[j+1] * A[j-1] od;
%p A132372     T[m] := [seq(coeff(C[2], x, j), j = 0..k)];
%p A132372     m := m + 1
%p A132372 od; ListTools:-Flatten(T) end:
%p A132372 A132372(10);  # _Peter Luschny_, Nov 16 2023
%Y A132372 Cf. A006318, A103136 (signed version), A033878 (transpose).
%Y A132372 Cf. A155069, A321620.
%K A132372 easy,nonn,tabl
%O A132372 0,4
%A A132372 _Philippe Deléham_, Nov 20 2007