[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],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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