A136807 Hankel transform of double factorial numbers n!*2^n=A000165(n).

1, 4, 256, 589824, 86973087744, 1282470362637926400, 2723154477021188283432960000, 1133321924829207204666583887642624000000, 120746421332702772771144114237731253721340313600000000

Offset: 0

Author

Paul Barry, Jan 23 2008

Keywords

• easy
• nonn

Comments

By the properties of the Hankel transform, a(n)=2^(n(n+1))*A055209(n).

Also Hankel transform of A000354, A010844, A082032. - Philippe Deléham, Jan 23 2008

Links

• G. C. Greubel, Table of n, a(n) for n = 0..28
• Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Crossrefs

Cf. A000354, A010844, A082032, A055209.

Programs

Magma

[1] cat [(&*[(2*k)^(2*(n-k+1)): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

Mathematica

Table[Product[(2k)^(2(n-k+1)),{k,n}],{n,0,10}] (* Harvey P. Dale, Apr 11 2013 *)

PARI

for(n=0,10, print1(prod(k=1,n,(2*k)^(2*(n-k+1))), ", ")) \\ G. C. Greubel, Oct 14 2018

Formula

a(n) = Product_{k=1..n} (2k)^(2(n-k+1)).

a(n) ~ 2^((n+1)^2) * Pi^(n+1) * n^(n^2 + 2*n + 5/6) / (A^2 * exp(3*n^2/2 + 2*n - 1/6)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Feb 24 2019