%I #44 Jan 08 2017 11:27:20
%S 4382314,59821998476834,338197165389273486,17314015796594772560245514,
%T 145853326344012138627669357202,
%U 12936469013977571458378002436843685186,15931675838688077485749893663903436780403973163302
%N a(n) = (binomial(2 * prime(n + 3), prime(n + 3)) * A005259(prime(n + 3) - 1) - 2)/prime(n + 3)^5 for n >= 1.
%C Let p be a prime > 5. Binomial(2 * p, p) * A005259(p - 1) == 2 (mod p^5). So a(n) is an integer.
%H Seiichi Manyama, <a href="/A276323/b276323.txt">Table of n, a(n) for n = 1..88</a>
%H Julian Rosen, <a href="http://arxiv.org/abs/1608.06864">Periods, supercongruences, and their motivic lifts</a>, arXiv:1608.06864 [math.NT], 2016.
%e a(1) = (binomial(14, 7) * A005259(6) - 2)/7^5 = (3432 * 21460825 - 2)/7^5 = 4382314.
%t Table[(Binomial[2 Prime[n + 3], Prime[n + 3]] Sum[(Binomial[#, k] Binomial[# + k, k])^2, {k, 0, #}] &[Prime[n + 3] - 1] - 2)/Prime[n + 3]^5, {n, 7}] (* _Michael De Vlieger_, Aug 30 2016 *)
%o (Ruby)
%o require 'prime'
%o def C(n, r)
%o r = [r, n - r].min
%o return 1 if r == 0
%o return n if r == 1
%o numerator = (n - r + 1..n).to_a
%o denominator = (1..r).to_a
%o (2..r).each{|p|
%o pivot = denominator[p - 1]
%o if pivot > 1
%o offset = (n - r) % p
%o (p - 1).step(r - 1, p){|k|
%o numerator[k - offset] /= pivot
%o denominator[k] /= pivot
%o }
%o end
%o }
%o result = 1
%o (0..r - 1).each{|k|
%o result *= numerator[k] if numerator[k] > 1
%o }
%o return result
%o end
%o def A005259(n)
%o i = 0
%o a, b = 1, 5
%o ary = [1]
%o while i < n
%o i += 1
%o a, b = b, ((((34 * i + 51) * i + 27) * i + 5) * b - i ** 3 * a) / (i + 1) ** 3
%o ary << a
%o end
%o ary
%o end
%o def A276323(n)
%o p_ary = Prime.take(n + 3)[3..-1]
%o a = A005259(p_ary[-1] - 1)
%o ary = []
%o p_ary.each{|i|
%o j = C(2 * i, i) * a[i - 1] - 2
%o break if j % i ** 5 > 0
%o ary << j / i ** 5
%o }
%o ary
%o end
%Y Cf. A000984, A005259.
%K nonn
%O 1,1
%A _Seiichi Manyama_, Aug 30 2016