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A358492 Irregular triangle read by rows: T(n,k) is one half of the number of line segments of length 1 in the k-th antidiagonal of the Dyck path described in the n-th row of A237593.

%I A358492 #32 Dec 15 2022 13:43:56
%S A358492 1,1,1,1,2,1,1,2,1,2,2,1,1,1,3,1,1,3,2,1,1,1,3,2,1,1,1,3,3,1,1,1,1,4,
%T A358492 2,1,1,1,4,2,2,1,1,1,1,1,3,4,1,1,1,1,3,4,2,1,1,1,1,2,4,2,2,1,1,1,1,1,
%U A358492 3,5,2,1,1,1,1,1,1,3,5,2,1,1,1,1,1,3,5,2,2,1,1,1,1,1,1,1,5,4,2,1,1,1,1,1,1,5,4,2,2
%N A358492 Irregular triangle read by rows: T(n,k) is one half of the number of line segments of length 1 in the k-th antidiagonal of the Dyck path described in the n-th row of A237593.
%e A358492 Triangle begins (first 19 rows):
%e A358492   1;
%e A358492   1, 1;
%e A358492   1, 2;
%e A358492   1, 1, 2;
%e A358492   1, 2, 2;
%e A358492   1, 1, 1, 3;
%e A358492   1, 1, 3, 2;
%e A358492   1, 1, 1, 3, 2;
%e A358492   1, 1, 1, 3, 3;
%e A358492   1, 1, 1, 1, 4, 2;
%e A358492   1, 1, 1, 4, 2, 2;
%e A358492   1, 1, 1, 1, 1, 3, 4;
%e A358492   1, 1, 1, 1, 3, 4, 2;
%e A358492   1, 1, 1, 1, 2, 4, 2, 2;
%e A358492   1, 1, 1, 1, 1, 3, 5, 2;
%e A358492   1, 1, 1, 1, 1, 1, 3, 5, 2;
%e A358492   1, 1, 1, 1, 1, 3, 5, 2, 2;
%e A358492   1, 1, 1, 1, 1, 1, 1, 5, 4, 2;
%e A358492   1, 1, 1, 1, 1, 1, 5, 4, 2, 2;
%e A358492 ...
%e A358492 For n = 10 the 10th row of A237593 is [6, 2, 1, 1, 1, 1, 2, 6]. When that row is interpreted as a symmetric Dyck path in the fourth quadrant using 20 line segments of length 1 the Dyck path looks like this:
%e A358492 .
%e A358492                          |
%e A358492                          |
%e A358492                          |
%e A358492                          |
%e A358492                          |
%e A358492                       _ _|
%e A358492                     _|
%e A358492                   _|
%e A358492                  |
%e A358492       _ _ _ _ _ _|
%e A358492 .
%e A358492 The numbers of line segments of length 1 in the successive antidiagonals are respectively [2, 2, 2, 2, 8, 4] so the 10th row of triangle is [1, 1, 1, 1, 4, 2].
%Y A358492 Row sums give A000027.
%Y A358492 Row n has length A008619(n).
%Y A358492 Column 1 gives A000012.
%Y A358492 Cf. A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A339575.
%K A358492 nonn,tabf
%O A358492 1,5
%A A358492 _Omar E. Pol_, Nov 19 2022