[seqfan] More on A002973 from Karol A. Penson

[Karol PENSON]

Considerations below concern A002793, in conjunction with the posts of 
Richard Mathar  and Max Alekseyev.
  First, thanks to  Max for his  egf  of this sequence which is perfect 
and for Richard for an elaborate path analogy , see below.

This sequence appears  in the process of conversion of a certain type  
of Meijer's G function,
in Maple notation :  MeijerG([[],[n]],[[0,0],[]],x) to more known 
special functions, namely  Ei(1,x).
In order to conveniently represent  this conversion ,  the  introduction 
of two auxilliary functions
is practical:

1. define   d(m,n)=n!*(-1)^m*hypergeom([-m,n+1],[1],1)/m!, which for 
integer m,n assumes
     only integer values;
2. furthermore,  define R(M,x)= 
Ei(1,x)*(M/((M-1)!))*((M-1)!+sum(d(M-1,p)*x^p,p=1..M-1)) ;

Then the sequence of following conversions
  seq(simplify(-exp(x)*((M)!/(M+1))*simplify(convert((M+1)!*MeijerG([[],[M+1]],[[0,0],[]],x),StandardFunctions)-R(M+1,x))),M=1..8)
produces the sequence of the following polynomials:

1, 3+x, 11 + 8 x + x^2 , 50 + 58 x + 15 x^2  + x^3 ,


   274 + 444 x + 177 x^2 + 24 x^3  + x^4 ,


   1764 + 3708 x + 2016 x^2 + 416 x^3  + 35 x^4  + x ^5,


   13068 + 33984 x + 23544 x^2  + 6560 x^3  + 835 x^4  + 48 x^5  + x^6 ,


   109584 + 341136 x + 288360 x ^2 + 101560 x^3  + 17370 x^4  + 1506 x 
^5 + 63 x ^6 + x^7 .

This triangle is exactly the one produced by Richard Mathar by using  an 
insightful path analogy.
  I still do not know how to manipulate Max's egf
in order to obtain the above triangle.

These polynomials for x=1 wind up to:

seq(simplify(-exp(x)*((M)!/(M+1))*simplify(convert((M+1)!*MeijerG([[],[M+1]],[[0,0],[]],1),StandardFunctions)-R(M+1,1))),M=1..8)=

, which  apart the initial zero is precisely A002793(M)  !  .