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%I #10 Oct 29 2021 09:07:40
%S 1,1,2,1,4,5,1,8,13,14,1,16,34,41,42,1,32,89,122,131,132,1,64,233,365,
%T 417,428,429,1,128,610,1094,1341,1416,1429,1430,1,256,1597,3281,4334,
%U 4744,4846,4861,4862,1,512,4181,9842,14041,16016,16645,16778,16795
%N Triangle read by rows of number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2n steps with all values less than or equal to k.
%C T(n,k) is the number of different out-stack sequences of n elements to be pushed into a stack of size k. E.g. T(3,2) = 4 since the 4 possible out-stack sequences are 123, 132, 213, 231; 321 is not allowed since it requires a stack of size 3. - _Jianing Song_, Oct 28 2021
%F For 1<=k<=n, T(n, k) =A080934(n, k) =T(n, k-1)+A080936(n, k).
%e Rows start:
%e 1;
%e 1,2;
%e 1,4,5;
%e 1,8,13,14;
%e 1,16,34,41,42;
%e ...
%e T(3,2)=4 since the paths of length 2*3 (7 points) with all values less than or equal to 2 can take the routes 0101010, 0101210, 0121010 or 0121210, but not 0123210.
%Y Cf. A000108, A079214, A080934, A080936.
%K nonn,tabl
%O 1,3
%A _Henry Bottomley_, Feb 25 2003
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