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}}

{\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 + 4n + 6)/72.

{\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).