binomialn1

Binomialn1 is a term used in some mathematical discussions to denote a specific binomial coefficient where the lower index is reduced by one. In this sense, Binomialn1(n) is defined as the binomial coefficient C(n, n−1) for natural numbers n ≥ 1. By the standard identity C(n, n−1) = n, Binomialn1(n) yields the simple integer sequence 1, 2, 3, 4, …

Notationally, Binomialn1 emphasizes the idea of fixing the upper index and shifting the lower index upward by one unit from the symmetric counterpart. The construction relies on the binomial coefficient definition C(n, k) = n!/(k!(n−k)!). Because C(n, n−1) is equal to C(n, 1), Binomialn1(n) also expresses the symmetry of binomial coefficients: choosing n−1 items from a set of n is equivalent to choosing the one item to leave out.

Combinatorial interpretation follows naturally: Binomialn1(n) counts the number of ways to omit a single element from an n-element set, leaving n−1 elements chosen. This perspective highlights the linear growth of Binomialn1(n) with n, in contrast to other binomial values that exhibit more complex growth patterns.

General observations show that Binomialn1 is not a standard, widely used notation in literature. It serves mainly as an instructive example to illustrate how shifting the lower index in a binomial coefficient affects the resulting value. In most contexts, C(n, n−1) is understood as equal to C(n, 1) = n.

See also: Binomial coefficient, Binomial theorem, Combinatorics.