This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A364386 #18 Apr 16 2024 18:00:53 %S A364386 1,0,1,1,0,1,2,1,0,1,4,3,1,0,1,8,7,4,1,0,1,18,15,11,5,1,0,1,44,33,26, %T A364386 16,6,1,0,1,113,78,59,42,22,7,1,0,1,296,197,138,101,64,29,8,1,0,1,782, %U A364386 518,342,240,165,93,37,9,1,0,1,2076,1388,892,590,406,258,130,46,10,1,0,1 %N A364386 Triangle T(n,k) read by rows: the number of Motzkin paths of length n that have k nodes at their peak level, 1 <= k <= n+1. %H A364386 Alois P. Heinz, <a href="/A364386/b364386.txt">Rows n = 0..140, flattened</a> %F A364386 T(n,n) = 1. (All nodes on level 0, only H steps.) %F A364386 T(n,n-1) = 0. %F A364386 T(n,n-2) = 1. (steps UHHH...HHHD) %e A364386 Example for 9 paths of length n=4: UUDD (k=1 at level 2), UHHD (k=3 at level 1), UHDH (k=2 at level 1), UDUD (k=2 at level 1), UDHH (k=1 at level 1), HUHD (k=2 at level 1), HUDH (k=1 at level 1), HHUD (k=1 at level 1), HHHH (k=5 at level 0). So k=1 appears 4 times, k=2 3 times, k=3 once, k=4 never, k=5 once. %e A364386 The triangle starts: %e A364386 1 %e A364386 0, 1 %e A364386 1, 0, 1 %e A364386 2, 1, 0, 1 %e A364386 4, 3, 1, 0, 1 %e A364386 8, 7, 4, 1, 0, 1 %e A364386 18, 15, 11, 5, 1, 0, 1 %e A364386 44, 33, 26, 16, 6, 1, 0, 1 %e A364386 113, 78, 59, 42, 22, 7, 1, 0, 1 %e A364386 296, 197, 138, 101, 64, 29, 8, 1, 0, 1 %e A364386 782, 518, 342, 240, 165, 93, 37, 9, 1, 0, 1 %e A364386 2076, 1388, 892, 590, 406, 258, 130, 46, 10, 1, 0, 1 %e A364386 5538, 3747, 2401, 1522, 1005, 665, 388, 176, 56, 11, 1, 0, 1 %e A364386 14856, 10147, 6560, 4085, 2576, 1680, 1054, 564, 232, 67, 12, 1, 0, 1 %e A364386 ... %Y A364386 Cf. A001006 (row sums), A088457 (column k=1). %Y A364386 Cf. A152879 (equivalent for Dyck paths). %K A364386 nonn,tabl %O A364386 0,7 %A A364386 _R. J. Mathar_, Jul 21 2023