A020474 A Motzkin triangle: a(n,k), n >= 2, 2 <= k <= n, = number of complete, strictly subdiagonal staircase functions.

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

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<k, 0, T(n, k-1)+T(n-1, k-1)+T(n-2, k-1))) \\ Ralf Stephan
(Haskell)
a020474 n k = a020474_tabl !! (n-2) !! (k-2)
a020474_row n = a020474_tabl !! (n-2)
a020474_tabl = map fst $ iterate f ([1], [0, 1]) where
   f (us, vs) = (vs, scanl (+) 0 ws) where
     ws = zipWith (+) (us ++ [0]) vs
-- Reinhard Zumkeller, Jan 03 2013
(Sage)
@cached_function
def T(n, k):
    if k<0 or n<k: return 0
    if k==0: return 0^n
    return T(n, k-1) + T(n-1, k-1) + T(n-2, k-1)
for n in (0..8): print([T(n, k) for k in (0..n)]) # Peter Luschny, Jun 23 2015

Crossrefs

Main diagonal is A001006.

Other diagonals include A002026, A005322, A005323, A005324, A005325. Row sums are (essentially) A005043.

The triangle version of A062105 has the same recurrence with different initial conditions. - N. J. A. Sloane, Apr 11 2020

Sequence in context: A329790 A343649 A195581 * A135589 A244312 A158122

Adjacent sequences: A020471 A020472 A020473 * A020475 A020476 A020477

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Feb 04 2000

Status

approved