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%I #14 Sep 08 2022 08:45:32
%S 1,4,256,589824,86973087744,1282470362637926400,
%T 2723154477021188283432960000,
%U 1133321924829207204666583887642624000000,120746421332702772771144114237731253721340313600000000
%N Hankel transform of double factorial numbers n!*2^n=A000165(n).
%C By the properties of the Hankel transform, a(n)=2^(n(n+1))*A055209(n).
%C Also Hankel transform of A000354, A010844, A082032. - _Philippe Deléham_, Jan 23 2008
%H G. C. Greubel, <a href="/A136807/b136807.txt">Table of n, a(n) for n = 0..28</a>
%H Michael Z. Spivey and Laura L. Steil, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%F a(n) = Product_{k=1..n} (2k)^(2(n-k+1)).
%F a(n) ~ 2^((n+1)^2) * Pi^(n+1) * n^(n^2 + 2*n + 5/6) / (A^2 * exp(3*n^2/2 + 2*n - 1/6)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Feb 24 2019
%t Table[Product[(2k)^(2(n-k+1)),{k,n}],{n,0,10}] (* _Harvey P. Dale_, Apr 11 2013 *)
%o (PARI) for(n=0,10, print1(prod(k=1,n,(2*k)^(2*(n-k+1))), ", ")) \\ _G. C. Greubel_, Oct 14 2018
%o (Magma) [1] cat [(&*[(2*k)^(2*(n-k+1)): k in [1..n]]): n in [1..10]]; // _G. C. Greubel_, Oct 14 2018
%Y Cf. A000354, A010844, A082032, A055209.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jan 23 2008
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