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EULER TRIANGLE


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 a are multiplied by a power of the index k:
    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)

Hints
Property [1] follows from property [2].
Property [2] follows from the recursive definition of S(n).

Marco Corvi - 2019