The Schröder number is the number of lattice paths
in the Cartesian plane that start at (0, 0), end
at ,
contain no points above the line, and are composed only of steps (0,
1), (1, 0), and (1, 1), i.e., , , and , where is a nonnegative integer.
The diagrams illustrating the paths generating , , and are illustrated above.
for ,
where ,
and the first few values , , ... are 2, 6, 22, 90, ... (OEIS A006318).
The numbers
of decimal digits in for , 1, ... are 1, 7, 74, 761, 7650, 76548, 765543, 7655504,
... (OEIS A114472), where (OEIS A104178).
Bonin, J.; Shapiro, L.; and Simion, R. "Some -Analogues of the Schröder Numbers Arising from Combinatorial
Statistics on Lattice Paths." J. Stat. Planning Inference34,
35-55, 1993. https://doi.org/10.1016/0378-3758(93)90032-2.Moser,
L. and Zayachkowski, W. "Lattice Paths with Diagonal Steps." Scripta
Math.26, 223-229, 1963.Pergola, E. and Sulanke, R. A.
"Schröder Triangles, Paths, and Parallelogram Polyominoes." J.
Integer Sequences1, No. 98.1.7, 1998. https://cs.uwaterloo.ca/journals/JIS/PergolaSulanke/.Rogers,
D. G. "A Schröder Triangle: Three Combinatorial Problems." In
Combinatorial Mathematics V: Proceedings of the Fifth Australian Conference
(Ed. C. H. C. Little). New York: Springer-Verlag, pp. 175-196,
1977. https://doi.org/10.1007/BFb0069192.Rogers,
D. G. and Shapiro, L. W. "Some Correspondences Involving the Schröder
Numbers and Relations." In Combinatorial Mathematics: Proceedings of the
International Conference on Combinatorial Theory, Canberra, 1977 (Ed. D. A. Holton
and J. Seberry). New York: Springer-Verlag, pp. 267-274, 1978. https://doi.org/10.1007/BFb0062541.Schröder,
E. "Vier kombinatorische Probleme." Z. Math. Phys.15, 361-376,
1870.Sloane, N. J. A. Sequences A006318/M1659,
A104178, and A114472
in "The On-Line Encyclopedia of Integer Sequences."Stanley,
R. P. "Hipparchus, Plutarch, Schröder, and Hough." Amer. Math.
Monthly104, 344-350, 1997. https://doi.org/10.1080/00029890.1997.11990645.Sulanke,
R. A. "Bijective Recurrences Concerning Schröder Paths." Elec.
J. Combin.5, R47, 1998. https://doi.org/10.37236/1385.
is the number of lattice paths
in the Cartesian plane that start at (0, 0), end
at 
,
contain no points above the line 
, and are composed only of steps (0,
1), (1, 0), and (1, 1), i.e., 
, 
, and 
, where 
is a nonnegative integer.
The diagrams illustrating the paths generating 
, 
, and 
are illustrated above.
are given by the recurrence relation
,
where 
,
and the first few values 
, 
, ... are 2, 6, 22, 90, ... (OEIS A006318).
The numbers 
of decimal digits in 
for 
, 1, ... are 1, 7, 74, 761, 7650, 76548, 765543, 7655504,
... (OEIS A114472), where 
(OEIS A104178).
is defined by continuity. It satisfies
![(1-x)G(x)-x[G(x)]^2=1.](/images/equations/SchroederNumber/NumberedEquation3.svg)
![1/2[-P_(n-1)(3)+6P_n(3)-P_(n+1)(3)],](/images/equations/SchroederNumber/Inline30.svg)
is a hypergeometric function, 
is a Gegenbauer
polynomial, and 
is a Legendre polynomial.
-Analogues of the Schröder Numbers Arising from Combinatorial
Statistics on Lattice Paths." J. Stat. Planning Inference 34,
35-55, 1993.