How to find a pattern in:

1, 3, 240, 840, 80640, 887040,...

What is the pattern in these polynomials (how to generate them)?

1. x^0

2. 3*x^2 + 3* x + 1

3. 240*x^4 + 480*x^3 + 380*x^2 + 140*x + 23

4. 840*x^6+2520*x^5+3220*x^4+2240*x^3+903*x^2+203*x+22

5. 80640*x^8+322560*x^7+571200*x^6+584640*x^5+379008*x^4+159936*x^3+43272*x^2+6984*x+563

sum((1+1/n)^n-exp(1), n = 1 .. infinity)

how to approach this problem: for what value of a, the expression is minimum:- g(n,a):=(1+1/(n))^(((n^(1/(a))+(n+1)^(1/(a) ))/(2))^(a))-exp(1) the answer seems to a=3.

is this identity true (i believe it is)?

p*int(f,x=0..infinity) = sum ((int(f,x=n..n+p)),n=0..n+p) + sum((p-n)*(int(f,x=n-1..n)),n=1...p-1)

p*(int(f(x), x = 0 .. infinity)) = sum(int(f(x), x = n .. n+p), n = 0 .. n+p)+sum((p-n)*(int(f(x), x = n-1 .. n)), n = 1 .. p-1)

If it is true how we understand it, and understand it through Maple.