%I #10 Apr 05 2020 09:34:02

%S 1,-1,-4,9,64,-225,-2304,11025,147456,-893025,-14745600,108056025,

%T 2123366400,-18261468225,-416179814400,4108830350625,106542032486400,

%U -1187451971330625,-34519618525593600,428670161650355625,13807847410237440000,-189043541287806830625,-6682998146554920960000

%N Row 2 of array in A288580.

%D F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.

%H J. Dezert, ed., <a href="http://www.mathematicsmagazine.com/corresp/J_Dezert/JDezert.htm">Smarandacheials (1)</a>, Mathematics Magazine for Grades 1-12, No. 4, 2004.

%H J. Dezert, ed., <a href="http://www.mathematicsmagazine.com/corresp/J_Dezert/JDezert2.htm">Smarandacheials (2)</a>, Mathematics Magazine for Grades 1-12, No. 4, 2004.

%F a(n) = !n!_2 = Prod_{i=0, 1, 2, ... .}_{0<|n-2i|<=n}(n-2i) = n(n-2)(n-4)... = (-1)^(Int(n+1)/2) * (n!!)^2

%e !3!_2 = 3(3-2)(3-4)(3-6) = 9 and !4!_2 = 4(4-2)(4-6)(4-8) = 64.

%p T:=proc(n,k) local i,p;

%p p:=1;

%p for i from 0 to floor(2*n/k) do

%p if n-k*i <> 0 then p:=p*(n-k*i) fi; od:

%p p;

%p end;

%p r:=k->[seq(T(n,k), n=0..60)]; r(2); # _N. J. A. Sloane_, Jul 03 2017

%t T[n_, k_] := Module[{i, p = 1}, For[i = 0, i <= Floor[2n/k], i++, If[n - k i != 0, p *= (n - k i)]]; p]; T[_, 0] = 1;

%t Table[T[n, 2], {n, 0, 22}] (* _Jean-François Alcover_, Apr 05 2020, after Maple *)

%Y Cf. A288580, A001818.

%K sign

%O 0,3

%A J. Dezert (Jean.Dezert(AT)onera.fr), Mar 21 2004

%E Entry revised by _N. J. A. Sloane_, Jul 03 2017