A092396 Row 2 of array in A288580.

1, -1, -4, 9, 64, -225, -2304, 11025, 147456, -893025, -14745600, 108056025, 2123366400, -18261468225, -416179814400, 4108830350625, 106542032486400, -1187451971330625, -34519618525593600, 428670161650355625, 13807847410237440000, -189043541287806830625, -6682998146554920960000

Offset: 0

Author

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

Keywords

• sign

Examples

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

References

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

Links

• J. Dezert, ed., Smarandacheials (1), Mathematics Magazine for Grades 1-12, No. 4, 2004.
• J. Dezert, ed., Smarandacheials (2), Mathematics Magazine for Grades 1-12, No. 4, 2004.

Crossrefs

Cf. A288580, A001818.

Programs

Maple

T:=proc(n,k)  local i,p;
p:=1;
for i from 0 to floor(2*n/k) do
if n-k*i <> 0 then p:=p*(n-k*i) fi; od:
p;
end;
r:=k->[seq(T(n,k), n=0..60)]; r(2); # N. J. A. Sloane, Jul 03 2017

Mathematica

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;
Table[T[n, 2], {n, 0, 22}] (* Jean-François Alcover, Apr 05 2020, after Maple *)

Formula

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

Extensions

Entry revised by N. J. A. Sloane, Jul 03 2017