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A108400
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a(n) = Product_{k = 0..n} (2^k * k!).
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9
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1, 2, 16, 768, 294912, 1132462080, 52183852646400, 33664847019245568000, 347485857744891213250560000, 64560982045934655213753964953600000, 239901585047846581083822477336190648320000000
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OFFSET
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0,2
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COMMENTS
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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
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
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LINKS
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M. E. Larsen, Wronskian Harmony, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37.
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FORMULA
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a(n) = Product_{i=1..n} Product_{j=0..i-1} {2*(i-j)}. - Paul Barry, Aug 02 2008
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
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MATHEMATICA
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Table[Product[k!*2^k, {k, 0, n}], {n, 0, 10}] (* Vaclav Kotesovec, Nov 14 2014 *)
Table[2^Binomial[n+1, 2]*BarnesG[n+2], {n, 0, 15}] (* G. C. Greubel, Jun 21 2022 *)
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PROG
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(Magma)
BarnesG:= func< n | (&*[Factorial(j): j in [0..n-2]]) >;
[2^Binomial(n+1, 2)*BarnesG(n+2): n in [0..15]]; // G. C. Greubel, Jun 21 2022
(SageMath)
def barnes_g(n): return product(factorial(j) for j in (0..n-2))
[2^binomial(n+1, 2)*barnes_g(n+2) for n in (0..15)] # G. C. Greubel, Jun 21 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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