

A211600


a(n) = (binomial(p^n, p^(n1))  binomial(p^(n1), p^(n2))) / p^(3n3) for p = 2.


2




OFFSET

3,2


COMMENTS

Consider the difference between two binomials f(p,k) = binomial(p^k, p^(k1))  binomial(p^(k1), p^(k2)).
A theorem from the A. I. Shirshov paper (in Russian) states:
p^(3k  3) divides f(p,k) for prime p = 2 and k > 2.
p^(3k  2) divides f(p,k) for prime p = 3 and k > 1.
p^(3k  1) divides f(p,k) for prime p > 3 and k > 1.


REFERENCES

D. B. Fuks and Serge Tabachnikov, Mathematical Omnibus: Thirty Lectures on Classic Mathematics, American Mathematical Society, 2007. Lecture 2. Arithmetical Properties of Binomial Coefficients, pages 2744


LINKS

Table of n, a(n) for n=3..8.
D. B. Fuks and M. B. Fuks, Arithmetics of binomial coefficients, Kvant 6 (1970), 1725. (in Russian)
A. I. Shirshov, On one property of binomial coefficients, Kvant 10 (1971), 1620. (in Russian)


FORMULA

a(n) = (binomial(2^n, 2^(n1)  binomial(2^(n1), 2^(n2))) / 2^(3n3).
a(n) = (A037293(n)  A037293(n1)) / 2^(3n  3).


EXAMPLE

a(3) = 1 is the difference between central binomials C(8,4)  C(4,2) = 70  6 = 64 divided by 2^(3*2  3) = 64.


MAPLE

A211600:=n>(binomial(2^n, 2^(n  1))  binomial(2^(n  1), 2^(n  2))) / 2^(3*n  3): seq(A211600(n), n=3..9); # Wesley Ivan Hurt, Apr 25 2017


MATHEMATICA

p = 2; Table[(Binomial[p^n, p^(n  1)]  Binomial[p^(n  1), p^(n  2)]) / 2^(3n  3), {n, 3, 9}]


CROSSREFS

Cf. A037293, A211601, A211602.
Sequence in context: A325215 A076445 A013835 * A068737 A151649 A122500
Adjacent sequences: A211597 A211598 A211599 * A211601 A211602 A211603


KEYWORD

nonn,easy


AUTHOR

Alexander Adamchuk, Apr 16 2012


STATUS

approved



