A211601 a(n) = (binomial(p^n, p^(n-1)) - binomial(p^(n-1), p^(n-2))) / p^(3n-2) for p = 3.

1, 2143, 39057044954221855, 507249004999029430448035076427591041390649615630234312261967

Offset

2,2

Comments

Consider the difference between two binomials f(p,k) = binomial(p^k, p^(k-1)) - binomial(p^(k-1), p^(k-2)).

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 27-44

Links

Table of n, a(n) for n=2..5.

D. B. Fuks and M. B. Fuks, Arithmetics of binomial coefficients, Kvant 6 (1970), 17-25. (in Russian)

A. I. Shirshov, On one property of binomial coefficients, Kvant 10 (1971), 16-20. (in Russian)

Formula

a(n) = ((binomial(3^n, 3^(n-1)) - binomial(3^(n-1), 3^(n-2))) / 3^(3n-2).

Mathematica

p = 3; Table[(Binomial[p^n, p^(n - 1)] - Binomial[p^(n - 1), p^(n - 2)]) / 3^(3n - 2), {n, 2, 6}]

Crossrefs

Cf. A211600, A211602.

Sequence in context: A116095 A347040 A252547 * A199745 A200145 A179271

Adjacent sequences: A211598 A211599 A211600 * A211602 A211603 A211604

Keyword

nonn

Author

Alexander Adamchuk, Apr 16 2012

Status

approved