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A204849 A Motzkin triangle by rows.

%I A204849 #22 Mar 03 2024 09:38:24
%S A204849 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 A204849 54,30,12,7,1,1,323,232,142,75,39,14,8,1,1,835,603,370,205,99,49,16,9,
%U A204849 1,1,2188,1585,983,545,281,126,60,18,10,1,1
%N A204849 A Motzkin triangle by rows.
%C A204849 Left border = A001006, row sums = A001006 with offset 1.
%C A204849 From _R. J. Mathar_, Jul 24 2017: (Start)
%C A204849 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 A204849 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 A204849 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 A204849 1, 1, 0, 0, 0, 0, 0, ...
%F A204849 1, 0, 1, 0, 0, 0, 0, ...
%F A204849 1, 1, 0, 1, 0, 0, 0, ...
%F A204849 1, 1, 1, 0, 1, 0, 0, ...
%F A204849 1, 1, 1, 1, 0, 1, 0, ...
%F A204849 1, 1, 1, 1, 1, 0, 1, ...
%F A204849 ...
%e A204849 First few rows of the triangle =
%e A204849      1;
%e A204849      1,    1;
%e A204849      2,    1,   1;
%e A204849      4,    3,   1,   1;
%e A204849      9,    6,   4,   1,   1;
%e A204849     21,   15,   8,   5,   1,   1;
%e A204849     51,   36,  22,  10,   6,   1,  1;
%e A204849    127,   91,  54,  30,  12,   7,  1,  1;
%e A204849    323,  232, 142,  75,  39,  14,  8,  1,  1;
%e A204849    835,  603, 370, 205,  99,  49, 16,  9,  1,  1;
%e A204849   2188, 1585, 983, 545, 281, 126, 60, 18, 10,  1, 1;
%e A204849   ...
%e A204849 Top row of M^3 = [4, 3, 1, 1, 0, 0, 0, ...].
%Y A204849 Cf. A001006.
%K A204849 nonn,tabl
%O A204849 0,4
%A A204849 _Gary W. Adamson_, Jan 19 2012