Generating The Series

Question

This is related to an ongoing event. It involves generating the following series : http://oeis.org/A008826

The generating Function as given in the above link is :

A(n,x) = Sum{ Stirling2(n,k) * A(k,x) * x } for k : 0 to n-1

where stirling2(n,k) is : http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind

I have made a simple loop to calculate the small A[n,k] :

for(int k = 0;k<n;k++)
    a(n,x) += stirling(n,k)*a(k,x)*x;

where stirling(n,k) is calculated by

stirling(n,k) = (stirling(n-1,k)*k)+stirling(n-1,k-1)

i have set

stirling(0,0) = 1
stirling(i,0) = stirling(0,i) = 0
A(1,i) = 1

The values of the Stirling Numbers being generated are correct. But the values for A(n,x) being generated do not match http://oeis.org/A008826/table

The output I am receiving is :

1       1       1       1       1   
1       2       3       4       5   
4       14      30      52      80  
32      198     606     1364    2580    
436     4722    20568   60004   139380

Did I get the function wrong ? Why isn't my output matching the one given in the link ? Am I interpreting the GF incorrectly ?

EDIT

I was finally able to calculate A(n;x) for small as well as large numbers. My final recurrence relation has come down to :

F(n) = A(n)*B(n) + (nC2 * F(n-1))

Last step to solve the recurrence relation :)

Answer 1

You're on the right track; you are just interpreting the a(n,x) incorrectly. The a(n,x) are polynomials - and the numbers in (n-1)st row of the triangular array are the coefficients by x, x^2, ... , x^n in a(n,x).

For instance, the polynomial a(4,x) is x+13x^2+18x^3, corresponding to the third row (namely 1 13 18) of the triangular array.