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%I #34 Sep 08 2022 08:46:01
%S 1,1,9,270,18900,2551500,589396500,214540326000,115851776040000,
%T 88626608670600000,92614806060777000000,128364121200236922000000,
%U 230285233433225038068000000,523898906060586961604700000000,1485253398681764036149324500000000
%N a(n) = (2n)! * (n+1)! / 2^(2n).
%C a(n) is always an integer since a(n+1)/a(n) = n^3 + 7/2*n^2 + 7/2*n + 1 which is always an integer. [_D. S. McNeil_, Dec 22 2011]
%C To further follow the above comment from _D. S. McNeil_: a(n+1)/a(n) is given in A002414. [_Alexander R. Povolotsky_, Dec 23 2011]
%H Vincenzo Librandi, <a href="/A202689/b202689.txt">Table of n, a(n) for n = 0..100</a>
%H T. Piezas, <a href="http://sites.google.com/site/piezas001/004">Notes and conjectures on properties of polynomials, arising in "Construction Of Binomial Sums For π And Polylogarithmic Constants Inspired by BBP Formulas"</a> (by Boris Gourévitch, Jesús Guillera Goyanes) and also relevant to further unpublished follow-up work by J. Cullen, T. Piezas, J. Guillera, and B. Gourevitch.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F a(n) = (2n)!(n+1)! / 2^(2n).
%F a(n+1) = a(n)*(n^3 + 7/2*n^2 + 7/2*n + 1).
%F a(n+1) = a(n)*A002414(n+1) for n >= 0.
%t Table[(2n)!(n+1)!/2^(2n),{n,0,20}] (* _Vincenzo Librandi_, Feb 09 2012 *)
%o (PARI) a(n)=(2*n)!*(n+1)!>>(2*n) \\ _Charles R Greathouse IV_, Dec 23 2011
%o (Magma) [Factorial(2*n)*Factorial(n+1)/2^(2*n): n in [0..15]]; // _Vincenzo Librandi_, Feb 09 2012
%Y Cf. A002414.
%K nonn,easy
%O 0,3
%A _Alexander R. Povolotsky_, Dec 22 2011
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