This site is supported by donations to The OEIS Foundation.

# Binomial identities :

• In case you spot any related sequences or have comments, you can post them on my user_talk_page.

## A000984 The Central Binomial Coefficient :

Reviewing some OEIS binomial related sequences one notices the following form for certain p and q :

 {\begin{aligned}a(n)&=\left(-q^{2}\right)^{n}{\binom {\frac {p}{q}}{n}}=q^{2n}{\binom {n-1-{\frac {p}{q}}}{-1-{\frac {p}{q}}}}={\frac {q^{2n}}{n!}}\left(-{\frac {p}{q}}\right)_{n}={\frac {q^{n}}{n!}}\prod _{k=0}^{n-1}{qk-p}=q^{2n}C_{n}^{\{-{\frac {p}{2q}}\}}{(1)}=\left[x^{n}\right]\left(1-q^{2}x\right)^{\frac {p}{q}}\end{aligned}} By assigning p=1 and q=-2 the sequence a(n) is the central binomial coefficient and one obtains the following identity:

 {\begin{aligned}{\binom {2n}{n}}=\left(-4\right)^{n}{\binom {-{\frac {1}{2}}}{n}}=4^{n}{\binom {n-{\frac {1}{2}}}{-{\frac {1}{2}}}}={\frac {4^{n}}{n!}}\left({\frac {1}{2}}\right)_{n}={\frac {(-2)^{n}}{n!}}\prod _{k=0}^{n-1}{-2k-1}=4^{n}C_{n}^{\{{\frac {1}{4}}\}}{(1)}=\left[x^{n}\right]{\frac {1}{\sqrt {1-4x}}}\end{aligned}} ## The Binomial Coefficient and its square :

 ${\binom {k}{i}}={\binom {k}{i}}^{2}+2\sum _{j=1}^{i}{(-1)^{j}{\binom {k}{i-j}}{\binom {k}{i+j}}}$ • It is a little challenging writing the identity this way but the (-1)^j takes care of the sign.
Due to its origin it is more meaningful using the variables (i,j,k).
For the OEIS sequences it is common to replace k by n.
 Related Sequence Name A108958 Number of unordered pairs of distinct length-n binary words having the same number of 1's. A054563 a(n) = n*(n^2 - 1)*(n + 2)*(n^2 + 4*n + 6)/72.

## The Binomial Coefficient with offset and its square :

 {\begin{aligned}{\binom {n+k}{k}}&={\binom {n+k}{k}}^{2}+2\sum _{i=1}^{k}(-1)^{i}{{\binom {n+k}{k+i}}{\binom {n+k}{k-i}}}\;\;&\forall \;\;{\;k\in \mathbb {N} }\\&={\binom {n+k}{k}}^{2}-2{\binom {n+k}{k+1}}{\binom {n+k}{k-1}}\,_{3}F_{2}(1,1-k,1-n;2+k,2+n;-1)\;\;&\forall \;\;{\;k\in \mathbb {Q} }\end{aligned}} n k Related Sequence Name n 1 A000027 The positive integers. n 2 A000217 Triangular numbers: a(n) = binomial(n+1,2). n 3 A000292 Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3). n 4 A000332 Binomial coefficient binomial(n,4) n 5 A000389 Binomial coefficients C(n,5). n 6 A000579 Figurate numbers or binomial coefficients C(n,6). n 7 A000580 a(n) = binomial coefficient C(n,7). n 8 A000581 a(n) = binomial coefficient C(n,8). n 9 A000582 a(n) = binomial coefficient C(n,9). n 10 A001287 a(n) = binomial coefficient C(n,10). n 11 A001288 a(n) = binomial(n,11). n 12 A010965 a(n) = binomial(n,12). n 13 A010966 a(n) = binomial(n,13).

 n k Related Sequence Name 2n 1 A005408 The odd numbers: a(n) = 2*n + 1. 2n 2 A000384 Hexagonal numbers : a (n) = n*(2*n - 1) = C(2*n,2). 2n 4 A053134 Binomial coefficients C (2*n + 4, 4). 2n 6 A053135 Binomial coefficients C (2*n + 6, 6). 2n 8 A053137 Binomial coefficients C (2*n + 8, 8). 2n 10 A196789 Binomial coefficients C(2*n+10,10).