%I #22 Mar 03 2024 09:38:24
%S 1,1,1,2,1,1,4,3,1,1,9,6,4,1,1,21,15,8,5,1,1,51,36,22,10,6,1,1,127,91,
%T 54,30,12,7,1,1,323,232,142,75,39,14,8,1,1,835,603,370,205,99,49,16,9,
%U 1,1,2188,1585,983,545,281,126,60,18,10,1,1
%N A Motzkin triangle by rows.
%C Left border = A001006, row sums = A001006 with offset 1.
%C From _R. J. Mathar_, Jul 24 2017: (Start)
%C The element T(n-1,k) counts the RGS's in Arndt's bijection of Apr 17 2013 in A001006 which have length n and finish with the k-th largest possible rise in the last step (0, 2, 3, 4, 5, ..., 1 impossible).
%C Example with n=4: the four RGS's 0000, 0022, 0033 and 0222 finish with a rise of 0 [T(3,0)=4]; the three RGS's 0002, 0024, 0224 finish with a rise of 2 [T(3,1)=3]; the one RGS 0003 finishes with a rise of 3 [T(3,2)=1]; the one 0004 finishes with a rise of 4 [T(3,3)=1]. (End)
%F n-th row of the triangle is the top row of M^n (deleting the zeros), where M = the following infinite square production matrix:
%F 1, 1, 0, 0, 0, 0, 0, ...
%F 1, 0, 1, 0, 0, 0, 0, ...
%F 1, 1, 0, 1, 0, 0, 0, ...
%F 1, 1, 1, 0, 1, 0, 0, ...
%F 1, 1, 1, 1, 0, 1, 0, ...
%F 1, 1, 1, 1, 1, 0, 1, ...
%F ...
%e First few rows of the triangle =
%e 1;
%e 1, 1;
%e 2, 1, 1;
%e 4, 3, 1, 1;
%e 9, 6, 4, 1, 1;
%e 21, 15, 8, 5, 1, 1;
%e 51, 36, 22, 10, 6, 1, 1;
%e 127, 91, 54, 30, 12, 7, 1, 1;
%e 323, 232, 142, 75, 39, 14, 8, 1, 1;
%e 835, 603, 370, 205, 99, 49, 16, 9, 1, 1;
%e 2188, 1585, 983, 545, 281, 126, 60, 18, 10, 1, 1;
%e ...
%e Top row of M^3 = [4, 3, 1, 1, 0, 0, 0, ...].
%Y Cf. A001006.
%K nonn,tabl
%O 0,4
%A _Gary W. Adamson_, Jan 19 2012