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 A020474 A Motzkin triangle: a(n,k), n >= 2, 2 <= k <= n, = number of complete, strictly subdiagonal staircase functions. 14
 1, 0, 1, 0, 1, 2, 0, 0, 2, 4, 0, 0, 1, 5, 9, 0, 0, 0, 3, 12, 21, 0, 0, 0, 1, 9, 30, 51, 0, 0, 0, 0, 4, 25, 76, 127, 0, 0, 0, 0, 1, 14, 69, 196, 323, 0, 0, 0, 0, 0, 5, 44, 189, 512, 835, 0, 0, 0, 0, 0, 1, 20, 133, 518, 1353, 2188, 0, 0, 0, 0, 0, 0, 6, 70, 392, 1422, 3610, 5798, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,6 COMMENTS T(n,k) = number of Dyck n-paths that start UU, contain no DUDU and no subpath of the form UUPDD with P a nonempty Dyck path and whose terminal descent has length n-k+2. For example, T(5,4)=2 counts UUDUUDUDDD, UUUDDUUDDD (each ending with exactly n-k+2=3 Ds). - David Callan, Sep 25 2006 LINKS Reinhard Zumkeller, Rows n = 2..120 of triangle, flattened M. Aigner, Motzkin Numbers, Europ. J. Comb. 19 (1998), 663-675. R. De Castro, A. L. Ramírez and J. L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, arXiv preprint arXiv:1310.2449 [cs.DM], 2013. J. L. Chandon, J. LeMaire and J. Pouget, Denombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 61-80. R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301. Paul Peart and Wen-jin Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Appl. Math. 98 (2000), 255-263. FORMULA a(n,k) = a(n,k-1) + a(n-1,k-1) + a(n-2,k-1), n > k >= 2. EXAMPLE Triangle begins: 1 0, 1 0, 1, 2 0, 0, 2, 4 0, 0, 1, 5, 9 0, 0, 0, 3, 12, 21 0, 0, 0, 1, 9, 30, 51 0, 0, 0, 0, 4, 25, 76, 127 0, 0, 0, 0, 1, 14, 69, 196, 323 MAPLE M:=16; T:=Array(0..M, 0..M, 0); T[0, 0]:=1; T[1, 1]:=1; for i from 1 to M do T[i, 0]:=0; od: for n from 2 to M do for k from 1 to n do T[n, k]:= T[n, k-1]+T[n-1, k-1]+T[n-2, k-1]; od: od; rho:=n->[seq(T[n, k], k=0..n)]; for n from 0 to M do lprint(rho(n)); od: # N. J. A. Sloane, Apr 11 2020 MATHEMATICA a[2, 2]=1; a[n_, k_]/; Not[n>2 && 2<=k<=n] := 0; a[n_, k_]/; (n>2 && 2<=k<=n) := a[n, k] = a[n, k-1] + a[n-1, k-1] + a[n-2, k-1]; Table[a[n, k], {n, 2, 10}, {k, 2, n}] (* David Callan, Sep 25 2006 *) PROG (PARI) T(n, k)=if(n==0&&k==0, 1, if(n<=0||k<=0||n

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Last modified December 6 15:54 EST 2022. Contains 358644 sequences. (Running on oeis4.)