The Catalan Numbers. Part II¹

• Autorzy: Karol Pąk
• Języki publikacji: EN

Abstrakty

EN

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 formula [...] and satisfies the recurrence relation [...] Obviously, 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 that [...] where [...] and hence [...] MML identifier: CATALAN2, version: 7.8.03 4.75.958

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2006
• Tom: 14
• Numer: 4
• Strony: 153-159

Daty

wydano

2006-01-01

online

2008-06-13

Twórcy

autor

Karol Pąk

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

Bibliografia

• [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
• [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
• [3] Patrick Braselmann and Peter Koepke. Equivalences of inconsistency and Henkin models. Formalized Mathematics, 13(1):45-48, 2005.
• [4] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
• [5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
• [6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
• [7] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
• [8] Dorota Czestochowska and Adam Grabowski. Catalan numbers. Formalized Mathematics, 12(3):351-353, 2004.
• [9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
• [10] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471-475, 1990.
• [11] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.
• [12] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.
• [13] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83-86, 1993.
• [14] Library Committee of the Association of Mizar Users. Binary operations on numbers. To appear in Formalized Mathematics.
• [15] Karol Pak. Cardinal numbers and finite sets. Formalized Mathematics, 13(3):399-406, 2005.
• [16] Karol Pak. Stirling numbers of the second kind. Formalized Mathematics, 13(2):337-345, 2005.
• [17] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125-130, 1991.
• [18] Konrad Raczkowski and Andrzej Nedzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.
• [19] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.
• [20] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.
• [21] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.
• [22] Andrzej Trybulec. Function domains and Fránkel operator. Formalized Mathematics, 1(3):495-500, 1990.
• [23] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9-11, 1990.
• [24] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.
• [25] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
• [26] Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.
• [27] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
• [28] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

Identyfikatory

DOI

10.2478/v10037-006-0019-7