repost: fun with Stirling numbers

[vdmcc]

Christian,

since you mentioned transforms on triangular tables,
I thought I'd dig up an oldie:

-----Original Message-----
From: Wouter Meeussen <eu000949 at pophost.eunet.be>
To: math-fun at optima.CS.Arizona.EDU <math-fun at optima.CS.Arizona.EDU>
Date: Tuesday, December 29, 1998 7:43 PM
Subject: fun with Stirling numbers


>hi,
>
>this relation is well known :
>
>In[1]:=Sum[ StirlingS2[w,K] Product[n-k,{k,0,K-1}],{K,0,w}]
>Out[1]=
>     n
>+ 31 (-1+n) n
>+ 90 (-2+n) (-1+n) n
>+ 65 (-3+n) (-2+n) (-1+n) n
>+ 15 (-4+n) (-3+n) (-2+n) (-1+n) n
>+    (-5+n) (-4+n) (-3+n) (-2+n) (-1+n) n
>
>In[2]:=%//Expand
>Out[2]=n^6
>
>
>but did you know it could be turned "upside down" into :
>
>
>In[3]:=Sum[ Sum[StirlingS2[n,K+1],{n,2,w+1}] Product[n-k,{k,1,K}],{K,0,w}]
>Out[3]=
>    6
>+ 120 (-1+n)
>+ 423 (-2+n) (-1+n)
>+ 426 (-3+n) (-2+n) (-1+n)
>+ 156 (-4+n) (-3+n) (-2+n) (-1+n)
>+  22 (-5+n) (-4+n) (-3+n) (-2+n) (-1+n)
>+     (-6+n) (-5+n) (-4+n) (-3+n) (-2+n) (-1+n)
>
>In[4]:=%//Expand
>Out[4]=
>(n + n^2 + n^3 + n^4 + n^5 + n^6)
>
>who ever heard about the StirlingS2[n,m] -triangle summed down the columns
>instead of summed by rows (to give the Bell-numbers)?
>
>In[5]:=Table[ Table[  Sum[  StirlingS2[n,K+1],{n,2,w+1}]
,{K,0,w}],{w,0,6}]
>Out[5]=
>{{0},
> {1,1},
> {2,4,1},
> {3,11,7,1},
> {4,26,32,11,1},
> {5,57,122,76,16,1},
> {6,120,423,426,156,22,1}}
>
>weird !
>
>Dr. Wouter L. J. MEEUSSEN
>w.meeussen.vdmcc at vandemoortele.be
>eu000949 at pophost.eunet.be
>