Re: referencesfor these coefficients

Am 09.09.05 04:56 schrieb Michael Blanc:

> Here's a triangle which I find somewhat interesting. The first few rows are:
>
>
> 1
> 1 1
> 1 4 1
> 1 11 11 1
> 1 26 66 26 1
> 1 57 302 302 57 1
> 1 120 1191 2416 1191 120 1
>
> The element at the intersection of the i_th and j_th diagonals is equal to
> the linear combination of its predecessors on those diagonals, using the
> weights i and j respectively.

As Rainer already wrote, this is the "eulerian triangle", whose
coefficients can be computed recursively or even directly from
row & column-index. There is a tight relation to the binomial-
coefficients; and they are useful for the computation of sums
of powers of consecutive numbers (have it not at hand actually)

If I also recall right, Bill Gosper had extended that triangle
above the top to negative indexed rows in one of his famous
HakMem-items, and I had also a formula to compute them. Note,
that the sequence of entries in the negative rows are infinite.

>
> (You might say that this is a souped-up version of Pascal's triangle.)
>
> It is amusing that the sum of the elements of the n_th row is the factorial
> of n.
>
> The triangle is actually useful, it seems. For example, the series
>
> 1 + 8x + 27x^2 + 64x^3 + ... = (1 + 4x + x^2) / (1 - x)^4 for |x| <
> 1.
>
> The coefficients of the series are the k_th powers of successive integers,
> and the coefficients of the polynomial of degree k-1 in the numerator on the
> RHS come from the triangle's k_th row; the exponent on (1-x) in the
> denominator is k+1. (The power-series expansion of 1 / 1-x to a positive
> integral power uses a diagonal of Pascal's triangle.)
>
> I just stumbled on this today while experimenting. Where is it in the
> literature? Are there any identities implied which are deeper than
> superficial? (The part of about n!, for example.)
>
I don't recall things right at the moment, but I had a small article,
where I "celebrated" these numbers as useful coefficients for the
computation of powersums. If that is of interest, you could try with

http://www.uni-kassel.de/~helms/priv/math/potenzsummen/potenzsummen_1.htm

and

http://www.uni-kassel.de/~helms/priv/math/hypergeomreihen/hypergeometrischereihen.htm

which are in german, but I think,at least in the first one the formulae
are detailed enough to understand things.

Gottfried Helms

.

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