Mathematics Magazine for Grades 1-12  

                                                   Smarandacheials (2)                                       

 1, 2

Edited by J. Dezert

4) In the case k=4:

     !n!₄ =  ∏(n-4i) = n(n-4)(n-8)… .

              0<|n-4i|≤n        i=0, 1, 2, … .

      Thus !9!₄ = 9(9-4)(9-8)(9-12)(9-16) = 9(5)(1)(-3)(-7) = 945.

        The sequence is: -15, 144, 105, 1024, 945, -14400, -10395, -147456, -135135, 2822400, 2027025,...

5) In the case k=5:

      !n!₅ =  ∏(n-5i) = n(n-5)(n-10)… .

        0<|n-5i|≤n         i=0, 1, 2, … .  

      Thus !11!₅ = 11(11-5)(11-10)(11-15)(11-20) = 11(6)(1)(-4)(-9) = 2376.

       The sequence is: -24, -42, 336, 216, 2500, 2376, 4032, -52416, -33264, -562500, -532224,

-891072, 16039296, … .   

        More general: Let n>k≥1 be two integers and m≥1 another integer.  Then the generalized Smarandacheial is defined as:

         !n!m_k =  ∏(n-k·i)

              0<|n-k·i|≤m    i=0, 1, 2, … .

          For examples:

          !7!3₂ = 7(7-2)(7-4)(7-6)(7-8)(7-10) = 7(5)(3)(1)(-1)(-3) = 315.

          !7!9₂ = 7(7-2)(7-4)(7-6)(7-8)(7-10)(7-12)(7-14)(7-16) = 7(5)(3)(1)(-1)(-3)(-5)(-7)(-9) = -99225.

References: J. Dezert, editor, “Smarandacheials”, Mathematics Magazine, Aurora, Canada, No. 4/2004;

www.gallup.unm.edu/~smarandache/Smarandacheials.htm.