Some combinatorial identities containing central binomial coefficients or Catalan numbers*
نویسندگان
چکیده
In the article, by virtue of Maclaurin's expansions arcsine function and its square cubic, authors give a short proof sum formula series with coefficients containing reciprocals Catalan numbers;establish four formulas for finite sums ratio or product two central binomial numbers.The instant simplifies discussions in journal papers: College Math. J. 43 (2012), no. 2, 141–146; Amer. Monthly 121 (2014), 3, 267–267; 123 (2016), 4, 405–406; Elem. 71 109–121; Mathematics 5 (2017), Article 40, 31 pages.
منابع مشابه
Series with Central Binomial Coefficients, Catalan Numbers, and Harmonic Numbers
We present several generating functions for sequences involving the central binomial coefficients and the harmonic numbers. In particular, we obtain the generating functions for the sequences ( 2n n ) Hn, ( 2n n ) 1 nHn, ( 2n n ) 1 n+1Hn , and ( 2n n ) n m. The technique is based on a special Euler-type series transformation formula.
متن کاملAsymptotic Expansions of Central Binomial Coefficients and Catalan Numbers
We give a systematic view of the asymptotic expansion of two well-known sequences, the central binomial coefficients and the Catalan numbers. The main point is explanation of the nature of the best shift in variable n, in order to obtain “nice” asymptotic expansions. We also give a complete asymptotic expansion of partial sums of these sequences.
متن کاملCatalan Triangle Numbers and Binomial Coefficients
The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac–Moody algebras. We prove that any binomial coefficient can be written as weighted sums along rows of the Catalan triangle. The coefficients in the sums form a triangular array, which we call the alternating Jacobsthal triangle. We study various subs...
متن کامل-Catalan Numbers and Squarefree Binomial Coefficients
In this paper we consider the generalized Catalan numbers F (s, n) = 1 (s−1)n+1 ( sn n ) , which we call s-Catalan numbers. We find all natural numbers n such that for p prime, p divides F (p, n), q ≥ 1 and all distinct residues of F (p, n) (mod p), q = 1, 2. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. We also prove that ( ...
متن کاملBinomial Coefficients , Catalan Numbers and Lucas Quotients
Let p be an odd prime and let a, m ∈ Z with a > 0 and p ∤ m. In this paper we determine p a −1 k=0 2k k+d /m k mod p 2 for d = 0, 1; for example, p a −1 k=0 2k k m k ≡ m 2 − 4m p a + m 2 − 4m p a−1 u p−(m 2 −4m p) (mod p 2), where (−) is the Jacobi symbol and {u n } n0 is the Lucas sequence given by u 0 = 0, u 1 = 1 and u n+1 = (m − 2)u n − u n−1 (n = 1, 2, 3,. . .). As an application, we deter...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Applied mathematics in science and engineering
سال: 2023
ISSN: ['2769-0911']
DOI: https://doi.org/10.1080/27690911.2023.2204233