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On the one hand, the recurrence relation uniquely determines the Catalan numbers; on the other hand, interpreting xc 2 − c + 1 = 0 as a quadratic equation of c and using the quadratic formula, the generating function relation can be algebraically solved to yield two solution possibilities
Whilst the above is a concrete example Catalan numbers, similar problems can be evaluated using Fuss-Catalan formula: Computer Stack: ways of arranging and completing a computer stack of instructions, each time step 1 instruction is processed and p new instructions arrive randomly. If at the beginning of the sequence there are r instructions ...
Cassini's formula was discovered in 1680 by Giovanni Domenico Cassini, then director of the Paris Observatory, and independently proven by Robert Simson (1753). [1] However Johannes Kepler presumably knew the identity already in 1608. [2] Catalan's identity is named after Eugène Catalan (1814–1894). It can be found in one of his private ...
Catalan's trapezoids are a countable set of number trapezoids which generalize Catalan’s triangle. Catalan's trapezoid of order m = 1, 2, 3, ... is a number trapezoid whose entries (,) give the number of strings consisting of n X-s and k Y-s such that in every initial segment of the string the number of Y-s does not exceed the number of X-s by m or more. [6]
6.2.1 Example: Generating function for the Catalan numbers 6.2.2 Example: Spanning trees of fans and convolutions of convolutions 6.3 Implicit generating functions and the Lagrange inversion formula
Robinson-Schensted algorithm, giving a proof of Burnside's formula for the symmetric group. Conjugation of Young diagrams, giving a proof of a classical result on the number of certain integer partitions. Bijective proofs of the pentagonal number theorem. Bijective proofs of the formula for the Catalan numbers.
See Faulhaber's formula. ... , generating function of the Catalan numbers [3] = =, ... in constant time even when the series contains a large number of ...
In number theory, Catalan's constant appears in a conjectured formula for the asymptotic number of primes of the form + according to Hardy and Littlewood's Conjecture F. However, it is an unsolved problem (one of Landau's problems) whether there are even infinitely many primes of this form. [9]