A211600 -id:A211600 - cp's OEIS frontend

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

Original entry on oeis.org

1, 2143, 39057044954221855, 507249004999029430448035076427591041390649615630234312261967

Offset: 2

Alexander Adamchuk, Apr 16 2012

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.

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

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)

Cf. A211600, A211602.

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

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

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