This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A211601 #17 Oct 26 2024 23:00:15
%S A211601 1,2143,39057044954221855,
%T A211601 507249004999029430448035076427591041390649615630234312261967
%N A211601 a(n) = (binomial(p^n, p^(n-1)) - binomial(p^(n-1), p^(n-2))) / p^(3n-2) for p = 3.
%C A211601 Consider the difference between two binomials f(p,k) = binomial(p^k, p^(k-1)) - binomial(p^(k-1), p^(k-2)).
%C A211601 A theorem from the A. I. Shirshov paper (in Russian) states:
%C A211601 p^(3k - 3) divides f(p,k) for prime p = 2 and k > 2.
%C A211601 p^(3k - 2) divides f(p,k) for prime p = 3 and k > 1.
%C A211601 p^(3k - 1) divides f(p,k) for prime p > 3 and k > 1.
%D A211601 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
%H A211601 D. B. Fuks and M. B. Fuks, <a href="http://kvant.mccme.ru/1970/06/arifmetika_binomialnyh_koeffic.htm">Arithmetics of binomial coefficients</a>, Kvant 6 (1970), 17-25. (in Russian)
%H A211601 A. I. Shirshov, <a href="http://kvant.mccme.ru/1971/10/ob_odnom_svojstve_binomialnyh.htm">On one property of binomial coefficients</a>, Kvant 10 (1971), 16-20. (in Russian)
%F A211601 a(n) = (binomial(3^n, 3^(n-1)) - binomial(3^(n-1), 3^(n-2))) / 3^(3*n-2).
%t A211601 p = 3; Table[(Binomial[p^n, p^(n - 1)] - Binomial[p^(n - 1), p^(n - 2)]) / 3^(3n - 2), {n, 2, 6}]
%Y A211601 Cf. A211600, A211602.
%K A211601 nonn,easy
%O A211601 2,2
%A A211601 _Alexander Adamchuk_, Apr 16 2012