# [seqfan] Re: A124824 ("LambertW analog of the Bell numbers") : Some mo re information of its occurences available?

Paul D Hanna pauldhanna at juno.com
Sat Feb 8 17:21:06 CET 2014

```Gottfried,
The formula for the sequence
A124824(n) = (1/e)*Sum_{k>=0} k*(n+k)^(n-1)/k!arose naturally as a variant of the well-known formula:
Bell(n) = (1/e)*Sum_{k>=0} k^n/k!
when conformed to the following property of the LambertW function:
( -LambertW(-x)/x )^k = Sum_{n>=0} k*(n+k)^(n-1)*x^n/n!
I submitted the sequence since it seemed interesting, and
Vladeta Jovovic quickly saw the (now obvious) formula:
E.g.f.: A(x) = exp(L(x) - 1), where L(x) = -LambertW(-x)/x.

But alas I do not know of any references that mention these numbers.
Wish you the best of health,
Paul

---------- Original Message ----------
From: Gottfried Helms <helms at uni-kassel.de>
To: "M <SeqFanList>" <seqfan at list.seqfan.eu>
Subject: [seqfan] A124824 ("LambertW analog of the Bell numbers") : Some more information of its occurences available?
Date: Sat, 08 Feb 2014 11:19:25 +0100

Seqfans -

just rereading an older treatize of mine I arrive again
at a sequence of numbers, which is known by

A124824: LambertW analog of the Bell numbers:
a(n) = (1/e)*Sum_{k>=0} k*(n+k)^(n-1)/k! for n>0 with a(0)=1.

I've already seen this in 2010 when I wrote my treatize; looking
again and with a fresh view at it I'd like now to know where this
sequence occured and whether the authors could possibly provide
interesting references.

I'd dealt with the question of "tetration of matrices"
see http://go.helms-net.de/math/tetdocs/PascalMatrixTetrated.pdf
page 10 "Exponential of PP"

--------------------

Also I'd like to ask whether there's someone around who
would like to help me improve the article - I think there's
something more in this concept worth to get digged out, but due
to weak health I don't think I'll be able to put sufficient energy
into a serious revision and further development of that study.

Gottfried

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