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
G. C. Greubel, Table of n, a(n) for n = 0..38
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) = 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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Jul 02 2005
STATUS
approved