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A167859 a(n) = 4^n * Sum_{k=0..n} binomial(2*k, k)^2 / 4^k.

%I A167859 #21 Jun 07 2021 04:43:34
%S A167859 1,8,68,672,7588,93856,1229200,16695424,232418596,3293578784,
%T A167859 47309094672,686870685312,10059942413584,148412250014336,
%U A167859 2202990595617344,32873407393419776,492791264816231204
%N A167859 a(n) = 4^n * Sum_{k=0..n} binomial(2*k, k)^2 / 4^k.
%C A167859 Every a(n) from a((p-1)/2) to a(p-1) is divisible by prime p for p = {7, 47, 191, 383, 439, 1151, 1399, 2351, 2879, 3119, 3511, 3559, ...} = A167860, apparently a subset of primes of the form 8n+7 (A007522).
%C A167859 7^3 divides a(13) and 7^2 divides a(10)-a(13).
%C A167859 Every a(n) from a(kp-1 - (p-1)/2) to a(kp-1) is divisible by prime p from A167860.
%C A167859 Every a(n) from a((p^2-1)/2) to a(p^2-1) is divisible by prime p from A167860. For p=7 every a(n) from a((p^3-1)/2) to a(p^3-1) and from a((p^4-1)/2) to a(p^4-1)is divisible by p^2.
%H A167859 Vincenzo Librandi, <a href="/A167859/b167859.txt">Table of n, a(n) for n = 0..300</a>
%F A167859 Recurrence: n^2*a(n) = 4*(5*n^2 - 4*n + 1)*a(n-1) - 16*(2*n - 1)^2*a(n-2). - _Vaclav Kotesovec_, Oct 20 2012
%F A167859 a(n) ~ 2^(4*n+2)/(3*Pi*n). - _Vaclav Kotesovec_, Oct 20 2012
%F A167859 G.f.: 2*EllipticK(4*sqrt(x))/(Pi*(1-4*x)), where EllipticK is the complete elliptic integral of the first kind, using the Gradshteyn and Ryzhik convention, also used by Maple.  In the convention of Abramowitz and Stegun, used by Mathematica, this would be written as 2*K(16*x)/(Pi*(1-4*x)). - _Robert Israel_, Sep 21 2016
%p A167859 A167859 := proc(n)
%p A167859     add( (binomial(2*k,k)/2^k)^2,k=0..n) ;
%p A167859     4^n*% ;
%p A167859 end proc:
%p A167859 seq(A167859(n),n=0..20) ; # _R. J. Mathar_, Sep 21 2016
%t A167859 Table[4^n*Sum[Binomial[2*k,k]^2/4^k,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%o A167859 (PARI) a(n) = 4^n*sum(k=0,n, binomial(2*k,k)^2/4^k) \\ _Charles R Greathouse IV_, Sep 21 2016
%Y A167859 Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167860, A007522.
%K A167859 nonn,easy
%O A167859 0,2
%A A167859 _Alexander Adamchuk_, Nov 13 2009
%E A167859 More terms from _Sean A. Irvine_, Apr 14 2010
%E A167859 Further terms from _Jon E. Schoenfield_, May 09 2010