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A108400
a(n) = Product_{k = 0..n} (2^k * k!).
9
1, 2, 16, 768, 294912, 1132462080, 52183852646400, 33664847019245568000, 347485857744891213250560000, 64560982045934655213753964953600000, 239901585047846581083822477336190648320000000
OFFSET
0,2
COMMENTS
Hankel transform (see A001906 for definition) of the sequences A000898, A001861, A035009(with first term omitted), A047974, A067147(unsigned version), A083886.
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
LINKS
M. E. Larsen, Wronskian Harmony, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
FORMULA
a(n) = A006125(n+1)*A000178(n).
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
MATHEMATICA
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 *)
PROG
(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
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Jul 02 2005
STATUS
approved