A276323 a(n) = (binomial(2 * prime(n + 3), prime(n + 3)) * A005259(prime(n + 3) - 1) - 2)/prime(n + 3)^5 for n >= 1.

4382314, 59821998476834, 338197165389273486, 17314015796594772560245514, 145853326344012138627669357202, 12936469013977571458378002436843685186, 15931675838688077485749893663903436780403973163302

Offset: 1

Author

Seiichi Manyama, Aug 30 2016

Keywords

• nonn

Comments

Let p be a prime > 5. Binomial(2 * p, p) * A005259(p - 1) == 2 (mod p^5). So a(n) is an integer.

Examples

			a(1) = (binomial(14, 7) * A005259(6) - 2)/7^5 = (3432 * 21460825 - 2)/7^5 = 4382314.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..88
• Julian Rosen, Periods, supercongruences, and their motivic lifts, arXiv:1608.06864 [math.NT], 2016.

Crossrefs

Cf. A000984, A005259.

Programs

Mathematica

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 *)

Ruby

require 'prime'
def C(n, r)
  r = [r, n - r].min
  return 1 if r == 0
  return n if r == 1
  numerator = (n - r + 1..n).to_a
  denominator = (1..r).to_a
  (2..r).each{|p|
    pivot = denominator[p - 1]
    if pivot > 1
      offset = (n - r) % p
      (p - 1).step(r - 1, p){|k|
        numerator[k - offset] /= pivot
        denominator[k] /= pivot
      }
    end
  }
  result = 1
  (0..r - 1).each{|k|
    result *= numerator[k] if numerator[k] > 1
  }
  return result
end
def A005259(n)
  i = 0
  a, b = 1, 5
  ary = [1]
  while i < n
    i += 1
    a, b = b, ((((34 * i + 51) * i + 27) * i + 5) * b - i ** 3 * a) / (i + 1) ** 3
    ary << a
  end
  ary
end
def A276323(n)
  p_ary = Prime.take(n + 3)[3..-1]
  a = A005259(p_ary[-1] - 1)
  ary = []
  p_ary.each{|i|
    j = C(2 * i, i) * a[i - 1] - 2
    break if j % i ** 5 > 0
    ary << j / i ** 5
  }
  ary
end