Calculators
Large Schröder Number Calculator
Calculate exact large Schröder numbers for diagonal-constrained lattice paths, privately in your browser.
Count paths from (0, 0) to (n, n) using east, north, and diagonal steps that never rise above y = x. Enter an index from 0 to 300. Exact calculations stay in your browser.
Count lattice paths that stay below the diagonal.
The large Schröder number S(n) counts paths from (0, 0) to (n, n) using east (1, 0), north (0, 1), and diagonal (1, 1) steps that never rise above y = x. Diagonal steps on the boundary are allowed. The empty path gives S(0) = 1. The first values are 1, 2, 6, 22, 90, 394. The recurrence is S(n) = S(n − 1) + Σ S(k) × S(n − 1 − k), where k runs from 0 to n − 1. The first term accounts for an initial diagonal step; the sum decomposes other paths at their first return to the diagonal. These are large Schröder numbers: small Schröder numbers are half these values for n ≥ 1. Delannoy numbers allow paths on both sides of the diagonal; Catalan paths disallow diagonal steps. Enter an index from 0 to 300 using up to three digits. Leading zeros and surrounding whitespace are accepted; signs, decimals, separators and scientific notation are rejected. Browser BigInt arithmetic preserves every digit. Results show the exact value and decimal digit count. Inputs are not uploaded or stored.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is the large Schröder number for n = 3?
S(3) = 22. The sequence begins 1, 2, 6, 22, 90, 394, starting at index zero.
How do large and small Schröder numbers differ?
For n ≥ 1, the large Schröder number is twice the small Schröder number. Both sequences start with 1 at index zero. This calculator returns large numbers.
How do these differ from Delannoy numbers?
Delannoy paths have no diagonal boundary restriction. Here paths must remain on or below y = x.
Are results exact and private?
Yes. BigInt preserves every result digit for indices 0 to 300, entirely in your browser.