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) & = {(- q^{2})}^{n} (\binom{\frac{p}{q}}{n}) = q^{2 n} (\binom{n - 1 - \frac{p}{q}}{- 1 - \frac{p}{q}}) = \frac{q^{2 n}}{n!} {(- \frac{p}{q})}_{n} = \frac{q^{n}}{n!} \prod_{k = 0}^{n - 1} q k - p = q^{2 n} C_{n}^{{- \frac{p}{2 q}}} (1) = [x^{n}] {(1 - q^{2} x)}^{\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{2 n}{n}) = {(- 4)}^{n} (\binom{- \frac{1}{2}}{n}) = 4^{n} (\binom{n - \frac{1}{2}}{- \frac{1}{2}}) = \frac{4^{n}}{n!} {(\frac{1}{2})}_{n} = \frac{(- 2)^{n}}{n!} \prod_{k = 0}^{n - 1} - 2 k - 1 = 4^{n} C_{n}^{{\frac{1}{4}}} (1) = [x^{n}] \frac{1}{\sqrt{1 - 4 x}} \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 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 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).