Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 18th 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexCreated [[binomial theorem]], and added a relevant lemma to [[freshman's dream]].

   Created binomial theorem, and added a relevant lemma to freshman’s dream.

2. • CommentRowNumber2.
   • CommentAuthorColin Tan
   • CommentTimeApr 20th 2014
   • PermaLink
   Author: Colin Tan
   Format: MarkdownItexFor the Pascal triangle recurrence to determine the binomial coefficients $\binom{n}{k}$ where $n$ and $k$ are integers, do we also need, for n a positive integer, the boundary condition $\binom{n}{n} = 1$?

   For the Pascal triangle recurrence to determine the binomial coefficients $\binom{n}{k}$ where $n$ and $k$ are integers, do we also need, for n a positive integer, the boundary condition $\binom{n}{n} = 1$?

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 20th 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexOh yes, that's true, we do need another boundary condition to determine $\binom{x}{k}$, which can be expressed in any one of various ways, e.g., as $\binom{0}{k} = 0$ whenever $k \neq 0$. This is for both positive and negative integers $k$. Thanks.

   Oh yes, that’s true, we do need another boundary condition to determine $\binom{x}{k}$, which can be expressed in any one of various ways, e.g., as $\binom{0}{k} = 0$ whenever $k \neq 0$. This is for both positive and negative integers $k$. Thanks.