A011117 Triangle of numbers S(x,y) = number of lattice paths from (0,0) to (x,y) that use step set { (0,1), (1,0), (2,0), (3,0), ....} and never pass below y = x.

1, 1, 1, 1, 2, 3, 1, 3, 7, 11, 1, 4, 12, 28, 45, 1, 5, 18, 52, 121, 197, 1, 6, 25, 84, 237, 550, 903, 1, 7, 33, 125, 403, 1119, 2591, 4279, 1, 8, 42, 176, 630, 1976, 5424, 12536, 20793, 1, 9, 52, 238, 930, 3206, 9860, 26832, 61921, 103049, 1, 10, 63

Offset: 0

Author

Robert Sulanke (sulanke(AT)diamond.idbsu.edu)

Keywords

• nonn
• tabl

Comments

When seen as polynomials with descending coefficients: evaluations are A006318 (x=1), A001003 (x=2).

Triangular array in A104219 transposed. - Philippe Deléham, Mar 16 2005

Triangle T(n,k), 0 <= k <= n, defined by: T(0,0) = 1, T(n,k) = T(n-1,k) + Sum_{j=0..k-1} 2^j*T(n-1,k-1-j). - Philippe Deléham, Oct 10 2005

Examples

			Triangle starts:
[0] [1]
[1] [1, 1]
[2] [1, 2,  3]
[3] [1, 3,  7,  11]
[4] [1, 4, 12,  28,  45]
[5] [1, 5, 18,  52, 121,  197]
[6] [1, 6, 25,  84, 237,  550,  903]
[7] [1, 7, 33, 125, 403, 1119, 2591,  4279]
[8] [1, 8, 42, 176, 630, 1976, 5424, 12536, 20793]
[9] [1, 9, 52, 238, 930, 3206, 9860, 26832, 61921, 103049]
		

Links

• E. Barcucci, E. Pergola, R. Pinzani and S. Rinaldi, ECO method and hill-free generalized Motzkin paths, Séminaire Lotharingien de Combinatoire, B46b (2001), 14 pp.
• E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.

Crossrefs

Cf. A084938.

Right-hand columns show convolutions of little Schroeder numbers with themselves: A001003, A010683, A010736, A010849.

Programs

Mathematica

f[ x_, y_ ] := f[ x, y ] = Module[ {return}, If[ x == 0, return = 1, If[ y == x-1, return = 0, return = f[ x, y-1 ] + Sum[ f[ k, y ], {k, 0, x-1} ] ] ]; return ]; Do[ Print[ Table[ f[ k, j ], {k, 0, j} ] ], {j, 10, 0, -1} ]

Sage

def A011117_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)+sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [prec(n, n-k) for k in (0..n-1)]
for n in (1..9): print(A011117_row(n)) # Peter Luschny, Mar 16 2016

Formula

S(m, n) = ((n-m+1)/(n+1))*Sum_{i=0..m-1} 2^(m-i-1)*binomial(n+1, i+1)*binomial(m-1, i).

Another version of triangle [1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...] = 1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 11, 0, 1, 4, 12, 28, 45, 0, 1, ..., where DELTA is Deléham's operator defined in A084938.

G.f.: 2/(1 + uv - 2v + sqrt(1 - 6uv + u^2v^2)). - Emeric Deutsch, Dec 25 2003

Sum_{k = 0..n} T(n, k) = A006318(n), large Schroeder numbers. - Philippe Deléham, Jul 10 2004. (This is because T(n, k) = number of royal paths (A006318) of length n with exactly n-k Northeast steps lying on the line y=x. - David Callan, Aug 02 2004)

S(n,m) = ((n-m+1)/m)*Sum_{k=1..m} binomial(m,k)*binomial(n+k,k-1), n >= m > 1; S(n,0)=1; S(n,m)=0, n < m. See the corresponding formula for A104219. - Wolfdieter Lang, Mar 16 2009