The sum of the powers of *a* < 1 from 0 to infinity is
well known,

S(0) = Sum_0..inf a^k = 1/(1-a)

Consider the sums in which the powers of

S(n) = Sum_0..inf k^n a^k

These can be written as

S(n) = (a * d/da) S(n-1) = (a * d/da)^n S(0) = 1/(1-a)^(n+1) P_n(a)

with the polynomial P_n(a) = Sum_k p(n;k) a^k.

The coefficients of the polynomial P_n(a) can be arranged in a "triangle" (see A008292),

n k: 1 2 3 4 1 ------------------- 1 . . . 2 ---------------- 1 1 . . 3 ------------- 1 4 1 . 4 ---------- 1 11 11 1 . 5 ------- 1 26 66 26 1 6 ---- 1 57 302 302 57 1 7 - 1 120 1191 2416 1191 120 1 8 - ...

**Prove these properties:**

**1.** The sum of the numbers on row n-th is *n!* (*n* factorial).
For example, the sum of the numbers on the 6-th row is 720.

Sum_k p(n;k) = n!

**2.** The k-th element of row *n* is obtained recursively from the k-th
and (k+1)-st elements of row *n-1*

p(n;k) = (n-k+1) p(n-1;k-1) + k p(n-1;k) for k=1 .. n/2 or (n+1)/2 assume p(n;0) = p(n;n+1) = 0.

For example,

p(7;3) = 1191 = 5 * 57 + 3 * 302 = (7-3+1) * p(6;2) + 3 * p(6;4)

Property [1] follows from property [2].

Property [2] follows from the recursive definition of S(n).

Marco Corvi - 2019