Proof of The Formula
There are several ways of explaining why the formula
solves the combinatorial problems listed above. The first proof below uses a generating function. The other proofs are examples of bijective proofs; they involve literally counting a collection of some kind of object to arrive at the correct formula.
Read more about this topic: Catalan Number
Famous quotes containing the words proof of the, proof of, proof and/or formula:
“If we view our children as stupid, naughty, disturbed, or guilty of their misdeeds, they will learn to behold themselves as foolish, faulty, or shameful specimens of humanity. They will regard us as judges from whom they wish to hide, and they will interpret everything we say as further proof of their unworthiness. If we view them as innocent, or at least merely ignorant, they will gain understanding from their experiences, and they will continue to regard us as wise partners.”
—Polly Berrien Berends (20th century)
“The insatiable thirst for everything which lies beyond, and which life reveals, is the most living proof of our immortality.”
—Charles Baudelaire (1821–1867)
“To cease to admire is a proof of deterioration.”
—Charles Horton Cooley (1864–1929)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)