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

1, 2, 16, 768, 294912, 1132462080, 52183852646400, 33664847019245568000, 347485857744891213250560000, 64560982045934655213753964953600000, 239901585047846581083822477336190648320000000

Offset: 0

Author

Philippe Deléham, Jul 02 2005

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A000178, A006125, A074962.

Cf. A000898, A001861, A001906, A035009, A047974, A067147, A083886.

Programs

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

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 *)

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

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