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 A167859 a(n) = 4^n*Sum_{ k=0..n } ((binomial(2*k,k))^2)/4^k. 7
 1, 8, 68, 672, 7588, 93856, 1229200, 16695424, 232418596, 3293578784, 47309094672, 686870685312, 10059942413584, 148412250014336, 2202990595617344, 32873407393419776, 492791264816231204 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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). 7^3 divides a(13) and 7^2 divides a(10)-a(13). Every a(n) from a(kp-1 - (p-1)/2) to a(kp-1) is divisible by prime p from A167860. 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. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..300 FORMULA 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 a(n) ~ 2^(4*n+2)/(3*Pi*n). - Vaclav Kotesovec, Oct 20 2012 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 MAPLE A167859 := proc(n)     add( (binomial(2*k, k)/2^k)^2, k=0..n) ;     4^n*% ; end proc: seq(A167859(n), n=0..20) ; # R. J. Mathar, Sep 21 2016 MATHEMATICA Table[4^n*Sum[Binomial[2*k, k]^2/4^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 20 2012 *) PROG (PARI) a(n) = 4^n*sum(k=0, n, binomial(2*k, k)^2/4^k) \\ Charles R Greathouse IV, Sep 21 2016 CROSSREFS Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167860, A007522. Sequence in context: A279266 A054915 A073555 * A243246 A113357 A030992 Adjacent sequences:  A167856 A167857 A167858 * A167860 A167861 A167862 KEYWORD nonn,easy AUTHOR Alexander Adamchuk, Nov 13 2009 EXTENSIONS More terms from Sean A. Irvine, Apr 14 2010 Further terms from Jon E. Schoenfield, May 09 2010 STATUS approved

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Last modified May 31 19:40 EDT 2020. Contains 334748 sequences. (Running on oeis4.)