# [seqfan] More on A002973 from Karol A. Penson

Karol PENSON penson at lptl.jussieu.fr
Thu Jul 8 19:10:13 CEST 2010

```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)/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)  !  .

```

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