# [seqfan] Misfortune with adding comment to A112492

Gottfried Helms helms at uni-kassel.de
Sat Dec 31 02:07:01 CET 2011

```https://oeis.org/A112492

I didn't want to delete the already existing comment - so I left
it in the form-field. Now in the database it seems, as if the
two first parts of the comment were mine.
Could some editor please correct the problem?
And could possibly a hint, how to deal with *adding/expanding*

We have now
----------------------------
>From Gottfried Helms, Dec 11 2001. (Start)

This expansion is based on the partial fraction identity: 1/product(x+j,j=1..m)= (1 + sum(((-1)^j)*binomial(m,j)*x/(x+j),j=1..m))/m!, e.g., p. 37 of the Ch. Jordan reference.

Another version of this triangle (without a column of 1's) is A008969.

The column sequences are, for m=1..10: A000012 (powers of 1), A000225, A001240, A001241, A001242, A111886-A111888.

The triangle occurs as U-factor in the LDU-decomposition of the matrix M defined by m(r,c)=1/(1+r)^c (r,c,beginning at 0).

Then

a(r,c)= m(r,c) * (1+r)!^(c-r)

An explicite expansion based on this can be made by defining a "recursice harmonic number" (rhn). (This representation is just a heuristic pattern-interpretation, no analytic proof yet available).
(...)
-----------------------

where we should have

------------------------
This expansion is based on the partial fraction identity: 1/product(x+j,j=1..m)= (1 + sum(((-1)^j)*binomial(m,j)*x/(x+j),j=1..m))/m!, e.g., p. 37 of the Ch. Jordan reference.

Another version of this triangle (without a column of 1's) is A008969.

The column sequences are, for m=1..10: A000012 (powers of 1), A000225, A001240, A001241, A001242, A111886-A111888.

>From Gottfried Helms, Dec 11 2001. (Start)
The triangle occurs as U-factor in the LDU-decomposition of the matrix M defined by m(r,c)=1/(1+r)^c (r,c,beginning at 0).

Then

a(r,c)= m(r,c) * (1+r)!^(c-r)

An explicite expansion based on this can be made by defining a "recursice harmonic number" (rhn). (This representation is just a heuristic pattern-interpretation, no analytic proof yet available).
(...)
----------------------------

Thank you very much -

and a happy new year to all seq-fan-tastics on earth!

Gottfried Helms

```