<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">In this paper, we define sequence dominated by 0, in which every initial fragment contains more zeroes than ones. If <italic>n</italic> ≥ 2 · <italic>m</italic> and <italic>n</italic> > 0, then the number of sequences dominated by 0 the length <italic>n</italic> including <italic>m</italic> of ones, is given by the formula<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="forma.2006.14.issue-4_figure2.gif"></inline-graphic>and satisfies the recurrence relation<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="forma.2006.14.issue-4_figure3.gif"></inline-graphic>Obviously, if <italic>n</italic> = 2 · <italic>m</italic>, then we obtain the recurrence relation for the Catalan numbers (starting from 0)<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="forma.2006.14.issue-4_figure4.gif"></inline-graphic>Using the above recurrence relation we can see that<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="forma.2006.14.issue-4_figure5.gif"></inline-graphic>where <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="forma.2006.14.issue-4_figure6.gif"></inline-graphic> and hence<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="forma.2006.14.issue-4_figure7.gif"></inline-graphic>MML identifier: <bold>CATALAN2</bold>, version: <bold>7.8.03 4.75.958</bold></abstract>

In this paper, we define sequence dominated by 0, in which every initial fragment contains more zeroes than ones. If n ≥ 2 · m and n > 0, then the number of sequences dominated by 0 the length n including m of ones, is given by the formulaand satisfies the recurrence relationObviously, if n = 2 · m, then we obtain the recurrence relation for the Catalan numbers (starting from 0)Using the above recurrence relation we can see thatwhere  and henceMML identifier: CATALAN2, version: 7.8.03 4.75.958

The Catalan Numbers. Part II1

Institute of Mathematics, University of Białystok, Akademicka 2, 15-267 Białystok, Poland

Formalized Mathematics

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In this paper, we define sequence dominated by 0, in which every initial fragment contains more zeroes than ones. If n ≥ 2 · m and n > 0, then the number of...

The Catalan Numbers. Part II¹

Published Online: Jun 13, 2008

Page range: 153 - 159

DOI: https://doi.org/10.2478/v10037-006-0019-7

This content is open access.