Shapiro, Louis W. A short proof of an identity of Touchard’s concerning Catalan numbers. (English) Zbl 0337.05012 J. Comb. Theory, Ser. A 20, 375-376 (1976). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 18 Documents MSC: 05A19 Combinatorial identities, bijective combinatorics 05A10 Factorials, binomial coefficients, combinatorial functions 11B39 Fibonacci and Lucas numbers and polynomials and generalizations PDFBibTeX XMLCite \textit{L. W. Shapiro}, J. Comb. Theory, Ser. A 20, 375--376 (1976; Zbl 0337.05012) Full Text: DOI Online Encyclopedia of Integer Sequences: Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n, having k ddu’s [here u = (1,1) and d = (1,-1)]. References: [1] Izbicki, H., Über Unterbaumes eines Baumes, Monatshefte F. Math., 74, 56-62 (1970) · Zbl 0209.28102 [2] Motzkin, T., Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products, Bull. Amer. Math. Soc., 54, 352-360 (1948) · Zbl 0032.24607 [3] Riordan, J., A note on Catalan parentheses, Amer. Math. Monthly, 80, 904-906 (1973) · Zbl 0301.05003 [4] L. Shapiroin; L. Shapiroin · Zbl 0327.05013 [5] Touchard, J., Sur Certaines Équations Fontionnelles, (Proc. Int. Math. Congress, Toronto (1924), Vol. 1 (1928)), 465 [6] Yaglom, A.; Yaglom, I., (Challenging Mathematical Problems with Elementary Solutions, Vol. I (1964), Holden-Day: Holden-Day San Francisco) · Zbl 0123.24201 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.