What is series of factorial?

Answer 1

I have a sequence starting with:

{1},

{2, -1},

{9, -8, 1},

{64, -81, 24, -1},

{625, -1024, 486, -64, 1},

{7776, -15625, 10240, -2430, 160, -1}, {117649, -279936, 234375, -81920, 10935, -384, 1}, that the row sums is: {1, 1, 2, 6, 24, 120, 720, …, }, the same is: {0!, 1!, 2!, 3!, 4!, 5!, 6!, …, n!}. In the sequence every row is a serie of factorial and the row sum is the value of factorial. This sequence is applied by the Law of factorialThe Factorial of every natural number n is analized and arranged unique by order and is alternately equal to sums and subtractions of the n+1 products of: every factorial of a base (from n+1, among n+1 bases, to the last number 1), with every number (from number 1, anong n+1 numbers, to the last number 1). The Property Formula of Factorial: n! = (n+1)n -nn´n +(n-1)n´n(n-1)/2 -(n-2)n´n(n-1)(n-2)/2´3 +(n-3)n´n(n-1)(n-2)(n-3)/2´3´4+… …+(-1)n-4´5 n´n(n-1)(n-2)(n-3)/2´3´4 +(-1)n-3´4n´n(n-1)(n-2)/2´3 +(-1)n-23n´n(n-1)/2 +(-1)n-1´2n´n +(-1)n´1 Explanation The sequence: {1},

{2, -1},

{9, -8, 1},

{64, -81, 24, -1},

{625, -1024, 486, -64, 1},

{7776, -15625, 10240, -2430, 160, -1}, {117649, -279936, 234375, -81920, 10935, -384, 1}, that each number is a product of every number of sequence {1},

{1, 1},

{1, 2, 1},

{1, 3, 3, 1},

{1, 4, 6, 4, 1},

{1, 5, 10, 10, 5, 1}, {1, 6, 15, 20, 15, 6, 1}, (A…)

with every number of sequence {10},

{21, 11},

{32, 22 ,12},

{43, 33, 23, 13},

{54, 44, 34, 24, 14},

{65, 55, 45, 35, 25, 15},

{76, 66, 56, 46, 36, 26, 16}, (A…) and every number of sequence{1},

{1, -1},

{1, -1, 1},

{1, -1, 1, -1},

{1, -1, 1, -1, 1},

{1, -1, 1, -1, 1, -1}, {1, -1, 1, -1, 1, -1, 1}, (A…) one by one, and by order. It is the property of the law of factorial.