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

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

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) = 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

Crossrefs

Cf. A000178, A006125, A074962.

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

Sequence in context: A278087 A015188 A005118 * A186002 A013029 A012915

Adjacent sequences: A108397 A108398 A108399 * A108401 A108402 A108403

Keyword

nonn,easy

Author

Philippe Deléham, Jul 02 2005

Status

approved