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User:Gerry Martens

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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 :

By assigning p=1 and q=-2 the sequence a(n) is the central binomial coefficient and one obtains the following identity:

The Binomial Coefficient and its square :

  • 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.
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 :

n k Related
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
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).