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A108400 a(n) = Product_{k = 0..n} (2^k * k!).

%I A108400 #23 Jun 21 2022 18:21:01
%S A108400 1,2,16,768,294912,1132462080,52183852646400,33664847019245568000,
%T A108400 347485857744891213250560000,64560982045934655213753964953600000,
%U A108400 239901585047846581083822477336190648320000000
%N A108400 a(n) = Product_{k = 0..n} (2^k * k!).
%C A108400 Hankel transform (see A001906 for definition) of the sequences A000898, A001861, A035009(with first term omitted), A047974, A067147(unsigned version), A083886.
%C A108400 Hankel transform of the sequence with e.g.f. exp(x^2). Also (-1)^C(n+1,2)*a(n) is the Hankel transform of the sequence with e.g.f. exp(-x^2). - _Paul Barry_, Feb 12 2008
%C A108400 Let T(n,k) = (n+1)^k * (1+(-1)^(n-k))/2, then a(n) = det(T(i,j); 0<=i, j<=n). - _Paul Barry_, Feb 12 2008
%H A108400 G. C. Greubel, <a href="/A108400/b108400.txt">Table of n, a(n) for n = 0..38</a>
%H A108400 M. E. Larsen, <a href="http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/wronskian-harmony">Wronskian Harmony</a>, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37.
%H A108400 J. W. Layman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%F A108400 a(n) = A006125(n+1)*A000178(n).
%F A108400 a(n) = Product_{i=1..n} Product_{j=0..i-1} {2*(i-j)}. - _Paul Barry_, Aug 02 2008
%F A108400 a(n) ~ 2^((n+1)^2/2) * n^(n^2/2+n+5/12) * Pi^((n+1)/2) / (A * exp(3*n^2/4+n-1/12)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - _Vaclav Kotesovec_, Nov 14 2014
%t A108400 Table[Product[k!*2^k, {k,0,n}], {n,0,10}] (* _Vaclav Kotesovec_, Nov 14 2014 *)
%t A108400 Table[2^Binomial[n+1,2]*BarnesG[n+2], {n,0,15}] (* _G. C. Greubel_, Jun 21 2022 *)
%o A108400 (Magma)
%o A108400 BarnesG:= func< n | (&*[Factorial(j): j in [0..n-2]]) >;
%o A108400 [2^Binomial(n+1,2)*BarnesG(n+2): n in [0..15]]; // _G. C. Greubel_, Jun 21 2022
%o A108400 (SageMath)
%o A108400 def barnes_g(n): return product(factorial(j) for j in (0..n-2))
%o A108400 [2^binomial(n+1,2)*barnes_g(n+2) for n in (0..15)] # _G. C. Greubel_, Jun 21 2022
%Y A108400 Cf. A000178, A006125, A074962.
%Y A108400 Cf. A000898, A001861, A001906, A035009, A047974, A067147, A083886.
%K A108400 nonn,easy
%O A108400 0,2
%A A108400 _Philippe Deléham_, Jul 02 2005