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

%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